f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim

Size: px
Start display at page:

Download "f (Pijk ) V. may form the Riemann sum: . Definition. The triple integral of f over the rectangular box B is defined to f (x, y, z) dv = lim"

Transcription

1 Chapter 14 Multiple Integrals..1 Double Integrals, Iterated Integrals, Cross-sections.2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals.3 Area and Volume by Double Integration, Volume by Iterated Integrals, Volume between Two surfaces.4 Double Integrals in Polar Coordinates, More general Regions.5 Applications of Double Integrals, Volume and First Theorem of Pappus, Surface Area and Second Theorem of Pappus, Moments of Inertia.6 Triple Integrals, Iterated Triple Integrals.7 Integration in Cylindrical and Spherical Coordinates..8 Surface Area, Surface Area of Parametric Surfaces, Surfaces Area in Cylindrical Coordinates.9 Change of Variables in Multiple Integrals, Jacobian

2 Chapter 14 Multiple Integrals Triple Integral, Upper and Lower Limits 14.5 Projections onto coordinate planes Mass, Moments, Centroid, Moment of Inertia.

3 Let B = [a, b] [c, d] [r, s] be a rectangular solid, and f (x, y, z) be a continuous scalar function defined on B. Divide B into l m n smaller rectangular solids. Label each small rectangular solid by C ijk, where 1 i l, 1 j m and 1 k n. Inside each such C ijk, pick a point P ijk = (x ijk, y ijk, z ijk ). Denote the volume of C ijk by V. Then we may form the Riemann sum: l m n i=1 j=1 k=1 f (P ijk ) V.. Definition. The triple integral of f over the rectangular box B is defined l m to f (x, y, z) dv = lim f (Pijk ) V.. B n l,m,n i=1 j=1 k=1 Remark. The triple integral over non-rectangular region can be defined similarly as that of double integral by summing up the smaller rectangles inside the region in the Riemann sum, and then obtain the limit as triple integral by using the Riemann sum.

4 Remarks. In general, it is difficult to compute the triple integral of scalar function over a solid region in space. So one can use the same idea which work very well in double integral, to evaluate the triple integral via iterated integral. Of course, we need to more. understanding of the region first.

5 Fubini s Theorem for non-rectangular region. Let D be a solid region in space.if D = { (x, y, z) R 3 (x, y) R, and z min (x, y) z z max (x, y) for all (x, y) R }, where z max (x, y) and z min (x, y) are continuous functions defined in the region D in xy-plane. Let f (x, y, z) be a scalar function defined in the region D, ( zmax ) (x,y) then f (x, y, z) dv = f (x, y, z)dz da.. D R z min (x,y) Remark. (1). The region R in xy-plane is in fact the shadow of the D under the projection from R 3 onto xy-plane, or by simply forgetting the z-coordinate. The idea is to allow the point P(x, y, ) varies within the region D, and then draw a line through P perpendicular to xy-plane, which will enter the solid D when z reaches z min (x, y) and then exit the solid D when z reaches z max (x, y). (2). The triple integral has similar properties like the double integral, here we con t repeat in stating these properties. (3). It is not necessary to project the solid D onto xy-plane, one can project D onto xz-plane, or yz-plane, in these cases, we should use function y = y(x, z) or x = x(y, z) respectively.

6 Sketch. Sketch the region D along with its "shadow" R (vertical projection) in the xy-plane. Label the upper and lower bounding surfaces. of D and the upper and lower bounding curves of R.

7 Find the z-limits of integration. Draw a line M passing through a typical point (x, y) in the shadow R parallel to the z-axis. As z increases, M enters the solid region D at z = z min (x, y) = f 1 (x, y) and. leaves at z = z max (x, y) = f 2 (x, y). These are the z-limits of integration

8 Find the y-limits of integration. Draw a line L through (x, y) parallel to the y-axis. As y increases, L enters R at y = y min = g 1 (x) and leaves at. y = y max (x) = g 2 (x). These are the y-limits of integration.

9 Find the x-limits of integration. Choose x-limits that include all lines through R parallel to the y-axis (x = a and x = b in the preceding figure). These are the x-limits of integration. The integral is x=b y=g2 (x) z=f1 (x,y). x=a y=g 1 (x) z=f 2 (x,y) F(x, y, z) dz dy dx.. Follow similar procedures if you change the order of integration. The "shadow" of region D lies in the plane of the last two variables with. respect to which the iterated integration takes place.

10 Example. Evaluate the triple integral dxdydz I =, where the solid D (1 + x + y + z) 3 D is bounded by the planes x. + y + z = 1, x =, y =, z =. Solution. Let R = { (x, y ) x + y 1, x, y } be the projection image of D onto xy-plane. Then D = { (x, y, z) x 1, y 1 x, z 1 x y }. ( dxdydz 1 x y ) I = D (1 + x + y + z) R 3 = dz (1 + x + y + z) 3 dxdy [ ] 1 1 x y = R 2(1 + x + y + z) 2 dxdy z= = 1 [ 1 2 R (1 + x + y) 2 1 ] dxdy 4 = y dx 2 (1 + x + y) 2 dy 1 16 = = 1 2 ln

11 Example. Evaluate D x 2 + z 2 dv, where D is the solid region bounded by the paraboloid S : y = x 2 + z 2 and the plane. π : y = 4. Solution. First determine the intersection of these surfaces E and S, take any intersection point P(x, y, z), so x 2 + z 2 = y = 4, so the intersection is a circle C : { (x, 4, z) x 2 + z 2 = 2 2 } in the plane π. We project D onto xz-plane, then its shadow R is bounded the image C of C. For any point Q(x, y, z) in D, then (x,, z) is in R, so we have y bottom (x, z) = x 2 + z = 4 = y top (x, z), this means that the plane π lies above the paraboloid S over the D, the solid D = { (x, y, z) x 2 + z 2 4, x 2 + z 2 y 4 }. So ymin D x 2 + z 2 (x,z)=4 dv = x 2 + z 2 dy da xz = D y min (x,z)=x 2 +z 2 (4 x 2 z 2 ) 2π 2 x 2 + z 2 da xz = (4 r 2 ) r 2 rdrdθ = D 2 [ 4r 2π 4r 2 r 4 3 ] 2 dr = 2π 3 r5 = 128π 5 15.

12 Example. Determine the volume of the solid D bounded above by the. plane z = y + 2 and bounded below by the paraboloid z = x 2 + y 2. Solution. Let P(x, y, z) be any point in the intersection of the plane and the paraboloid, we have x 2 + y 2 = z = x + 2, i.e. x 2 + (y 1 2 )2 = 9 4. Then shadow R of D on xy-plane is given by { (x, y, ) x 2 + (y 1 2 )2 = 9 4 }. It follows that the volume of D is given by dv = (z top z bottom ) dzda D R = (y + 2 x 2 y 2 ) da. R Remark. We will complete the calculation in the following example.

13 Example. Determine the volume of the solid D bounded above by the. plane z = y + 2 and bounded below by the paraboloid z = x 2 + y 2. Solution. If we project the solid D onto yz-plane, then its shadow T in yz-plane is bounded by z = y 2 and the line z = y + 2. In this case, in yz-plane, we have T = { (y, z) 1 y 2, y 2 z y + 2 }. For any point P(, y, z) T, the line through P(, y, z), parallel to x-axis meet the paraboloid z = x 2 + y 2 at Q(x max, y, z) and Q (x max, y, z), so x max (y, z) = z y 2, and x min (y, z) = z y 2. The volume of D is xmax 2 y+2 z y 2 given by dv = dx da zy = dx dz dy D T x min 1 y 2 z y2 2 y+2 2 [ 2 = 2 z y 2 dzdy = 2 (z y 2) ] 3/2 y+2 dy 1 y y 2 = y y 3 1(2 2 ) 3/2 dy = 4 3/2 ( ) 9 3/2 3 3/2 4 u2 du = 27 π/2 cos 4 θ dθ = 81 4 π/2 32 π. Remarks. In (*) we use u = y 1 2, and in ( ), we use u = 3 2 sin θ.

14 Example. Find the volume of the solid region D enclosed by the surfaces. z = x 2 + 3y 2 and z = 8 x 2 y 2. Solution. To find the limits of integration for evaluating the integral, we first sketch the region D. The surfaces (as shown above) intersect on the elliptical cylinder x 2 + 3y 2 = 8 x 2 y 2 or x 2 + 2y 2 = 4, z >. The boundary of the shadow R of D, after the projection onto xy-plane, is an ellipse with the same equation: x 2 + 2y 2 = 4. The upper boundary of R is the curve y = (4 x 2 )/2. The lower boundary is the curve y = (4 x 2 )/2. In this case, one can see that 2 x 2. It remains to find the z-limits of integration. The line M passing through a typical point (x, y) in R parallel to the z-axis enters D at z = x 2 + 3y 2 and leaves at z = 8 x 2 y 2.

15 Example. Find the volume of the solid region D enclosed by the surfaces. z = x 2 + 3y 2 and z = 8 x 2 y 2. Solution. The volume of the solid D is 2 ymax = (4 x 2 )/2 dv = = = = = D 2 y min = (4 x 2 )/2 2 (4 x 2 )/ = (4 x 2 )/2 (8 2x 2 4y 2 ) dy dx [(8 2x 2 )y 43 y3 ] y= (4 x 2 )/2 y= (4 x 2 )/2 zmax =8 x 2 y 2 z min =x 2 +3y 2 dx ( 4 x 2(8 2x 2 2 ) 8 ( 4 x 2 ) 3/2 ) dx [ ( 4 x 2 ) 3/2 8 8 ( 4 x 2 ) 3/2 ] dx dz dy dx (4 x 2 ) 3/2 dx = 8π 2. (* after plugging x = 2 sin u.)

16 Example. Set up the limits of integration in dydzdx, for evaluating the triple integral of a function F(x, y, z) over the tetrahedron D with vertices. (,, ), (1, 1, ), (, 1, ), and (, 1, 1). Solution. We sketch D along with its "shadow" R in the xz-plane. The upper (right-hand) bounding surface of D lies in the plane y = 1. The lower (left-hand) bounding surface lies in the plane y = x + z. The upper boundary of R is the line z = 1 x. The lower boundary is the line z =. R = { (x, z) x 1, z 1 x }. It remains to find the y-limits of integration. A line through a typical point (x, z) in R parallel to the y- axis enters D at y = x + z and leaves at y = 1. The required integral is 1 1 x 1 F(x, y, z) dv = F(x, y, z) dy dz dx. D x+z

17 Example. Set up the limits of integration in dz dy, dx, for evaluating the triple integral of a function F(x, y, z) over the tetrahedron D with vertices. (,, ), (1, 1, ), (, 1, ), and (, 1, 1). The required integral is 1 F(x, y, z) dv = D 1 y x x F(x, y, z) dz dy dx.

18 Example. Determine the volume of the solid region D in the first octant bounded by the coordinate planes and the surface z. = 4 x 2 y Solution. The level surface z = 4 x 2 y intersects the xy-plane at a curve C : 4 = x 2 + y. It follows that the shadow R of D on the xy-plane is given by R = { (x, y) x 2, y 4 x 2 }. In this case, the top is given by z max (x, y) = 4 x 2 y. Then the volume of D is given by (4 x 2 y) da = = = x R 2 ] 4 x 2 (4 x 2 y) dy dz = [(4 x 2 )y y2 2 [(4 x 2 ) 2 (4 x2 ) 2 ] dx = 1 2 (4 x 2 ) 2 dx 2 2 (16 8x 2 + x 4 ) dx = 1 ( = = 16 ( ) = )

19 Example. Determine the volume of the solid region D in the first octant bounded by the coordinate planes, the plane x + y = 4, and the. cylinder y 2 + 4z 2 = 16. Solution. The shadow R of D on the yz-plane is given by R = { (y, z) x, y, and y 2 + 4z 2 16 } = { (y, z) z 2, y 2 4 z 2 }. In this case, the top is given by x max (y, z) = 4 y. Then the volume of D is given by z 2 (4 y) da = (4 y) dy dz = = R 2 (8 4 z 2 2(4 z 2 ) ) dz 2 = π = 8π ] 2 4 z 2 [4y y2 dz 2

20 Example. Given D dv = 1 as. an equivalent iterated integral in the order (a) dy dz dx y x 2 dzdydx, rewrite the integral Solution. For any point P(x,, ), the line through P parallel to y-axis meet the solid D at the curve y = x 2 at Q, so Q(x, x 2, ). Then the line through point Q parallel to z-axis meets the top y + z = 1 before leaving the solid D at R. Hence R(x, x 2, 1 x 2 ). The shadow S of D on the xz-plane is S = { (x,, z) 1 x 1, z 1 x 2 }. (Continue..)

21 Example. Given D dv = 1 as. an equivalent iterated integral in the order (a) dy dz dx y x 2 dzdydx, rewrite the integral Solution. For any point P(x, z, ) in S, the line through P first passes through the cylinder (side) y = x 2 before entering the solid D, and then meets the plane (top) y + z = 1 before leaving the solid D, it follows that y min = x 2, and y max = 1 z. Hence, we have 1 1 x 2 1 z dv = dy dz dx. D 1 x 2

22 Example. The figure shows the region of integration for the integral y x f (x, y, z) dz dy dx. Rewrite this integral as an equivalent iterated. integral in dy dz dx. Solution. For any point A(x,, ) on the x-axis, the line through A parallel to y-axis meets the cylinder y = x at a point B(x, x, ). Then the line through B parallel to z-axis meets the plane z = 1 y at C(x, x, 1 x). It follows that the shadow R of solid D onto xz-plane is given by R = { (x, z) x 1, z 1 x }. For any point P(x,, z) in the shadow, the line through P parallel to y-axis meets the cylinder y = x at the point (x, x, z) before entering the solid D and and meets the plane z = 1 y at the point (x, 1 z, z), it follows that y min = x, and y max = 1 z. It follows that D = { (x, y, z) x 1, z 1 x, x y 1 z }. Then we have 1 1 x 1 y f (x, y, z) dz dy dx = 1 1 x 1 z x dy dz dx.

23 Example. The figure shows the solid region D of integration for the integral 1 1 x 2 1 x f (x, y, z) dy dz dx. Rewrite this integral as an. equivalent iterated integral in the dz dy dx order. Solution. The shadow R of D onto xy-plane is the triangle given by R = { (x, y) x 1, y 1 x }. Note that the top of the solid D is the graph z = 1 x 2, and the bottom of D is xy-plane, i.e. z max (x, y) = 1 x 2, and z min (x, y) =. The desired iterated integral is 1 1 x 1 x 2 f (x, y, z) dz dy dx.

24 Example. The figure shows the solid region D of integration for the integral 1 1 x 2 1 x f (x, y, z) dy dz dx. Rewrite this integral as an. equivalent iterated integral in the dz dy dx order. Solution. For any P(x, y, z) in the intersection curve C (in red) of the two surfaces z = 1 x 2, and y = 1 x. Then P(x, y, z) satisfies the equations: z = 1 x 2, and y = 1 x. It follows that P(x, 1 x, 1 x 2 ). After projecting the point P(x, 1 x, 1 x 2 ) of the curve C onto yz-plane, we have Q(, 1 x, 1 x 2 ), set y = 1 x, and z = 1 x 2, i.e. z = (1 x)(1 + x) = (1 x)( 1 x) = y( 2 + y) = 2y y 2 for all y 1. Let R 1 = { (y, z) y 1, z y(2 + y) } and R 2 = { (y, z) y 1, y(2 + y) z 1 }. It follows that the integral is given by 1 2y y 2 1 z dx dz dy y y 2 1 y dx dz dy.

Chapter 15 Notes, Stewart 7e

Chapter 15 Notes, Stewart 7e Contents 15.2 Iterated Integrals..................................... 2 15.3 Double Integrals over General Regions......................... 5 15.4 Double Integrals in Polar Coordinates..........................

More information

Double Integrals, Iterated Integrals, Cross-sections

Double Integrals, Iterated Integrals, Cross-sections Chapter 14 Multiple Integrals 1 ouble Integrals, Iterated Integrals, Cross-sections 2 ouble Integrals over more general regions, efinition, Evaluation of ouble Integrals, Properties of ouble Integrals

More information

Applications of Triple Integrals

Applications of Triple Integrals Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals

More information

= f (a, b) + (hf x + kf y ) (a,b) +

= f (a, b) + (hf x + kf y ) (a,b) + Chapter 14 Multiple Integrals 1 Double Integrals, Iterated Integrals, Cross-sections 2 Double Integrals over more general regions, Definition, Evaluation of Double Integrals, Properties of Double Integrals

More information

12.5 Triple Integrals

12.5 Triple Integrals 1.5 Triple Integrals Arkansas Tech University MATH 94: Calculus III r. Marcel B Finan In Sections 1.1-1., we showed how a function of two variables can be integrated over a region in -space and how integration

More information

Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate.

Math 2130 Practice Problems Sec Name. Change the Cartesian integral to an equivalent polar integral, and then evaluate. Math 10 Practice Problems Sec 1.-1. Name Change the Cartesian integral to an equivalent polar integral, and then evaluate. 1) 5 5 - x dy dx -5 0 A) 5 B) C) 15 D) 5 ) 0 0-8 - 6 - x (8 + ln 9) A) 1 1 + x

More information

Math 11 Fall 2016 Section 1 Monday, October 17, 2016

Math 11 Fall 2016 Section 1 Monday, October 17, 2016 Math 11 Fall 16 Section 1 Monday, October 17, 16 First, some important points from the last class: f(x, y, z) dv, the integral (with respect to volume) of f over the three-dimensional region, is a triple

More information

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13

Contents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Homework 1 - Solutions 3. 2 Homework 2 - Solutions 13 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Homework - Solutions 3 2 Homework 2 - Solutions 3 3 Homework 3 - Solutions 9 MATH 32B-2 (8) (L) G. Liu / (TA) A. Zhou Calculus

More information

MATH 261 EXAM III PRACTICE PROBLEMS

MATH 261 EXAM III PRACTICE PROBLEMS MATH 6 EXAM III PRACTICE PROBLEMS These practice problems are pulled from actual midterms in previous semesters. Exam 3 typically has 5 (not 6!) problems on it, with no more than one problem of any given

More information

UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx

UNIVERSITI TEKNOLOGI MALAYSIA SSCE 1993 ENGINEERING MATHEMATICS II TUTORIAL 2. 1 x cos dy dx x y dy dx. y cosxdy dx UNIVESITI TEKNOLOI MALAYSIA SSCE 99 ENINEEIN MATHEMATICS II TUTOIAL. Evaluate the following iterated integrals. (e) (g) (i) x x x sinx x e x y dy dx x dy dx y y cosxdy dx xy x + dxdy (f) (h) (y + x)dy

More information

Triple Integrals in Rectangular Coordinates

Triple Integrals in Rectangular Coordinates Triple Integrals in Rectangular Coordinates P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Triple Integrals in Rectangular Coordinates April 10, 2017 1 / 28 Overview We use triple integrals

More information

Double Integrals over Polar Coordinate

Double Integrals over Polar Coordinate 1. 15.4 DOUBLE INTEGRALS OVER POLAR COORDINATE 1 15.4 Double Integrals over Polar Coordinate 1. Polar Coordinates. The polar coordinates (r, θ) of a point are related to the rectangular coordinates (x,y)

More information

Triple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals

Triple Integrals. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Triple Integrals Triple Integrals MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 211 Riemann Sum Approach Suppose we wish to integrate w f (x, y, z), a continuous function, on the box-shaped region

More information

Math 265 Exam 3 Solutions

Math 265 Exam 3 Solutions C Roettger, Fall 16 Math 265 Exam 3 Solutions Problem 1 Let D be the region inside the circle r 5 sin θ but outside the cardioid r 2 + sin θ. Find the area of D. Note that r and θ denote polar coordinates.

More information

2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0.

2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0. Midterm 3 Review Short Answer 2. Give an example of a non-constant function f(x, y) such that the average value of f over is 0. 3. Compute the Riemann sum for the double integral where for the given grid

More information

Triple Integrals. Be able to set up and evaluate triple integrals over rectangular boxes.

Triple Integrals. Be able to set up and evaluate triple integrals over rectangular boxes. SUGGESTED REFERENCE MATERIAL: Triple Integrals As you work through the problems listed below, you should reference Chapters 4.5 & 4.6 of the recommended textbook (or the equivalent chapter in your alternative

More information

Math Triple Integrals in Cylindrical Coordinates

Math Triple Integrals in Cylindrical Coordinates Math 213 - Triple Integrals in Cylindrical Coordinates Peter A. Perry University of Kentucky November 2, 218 Homework Re-read section 15.7 Work on section 15.7, problems 1-13 (odd), 17-21 (odd) from Stewart

More information

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions

Calculus III. Math 233 Spring In-term exam April 11th. Suggested solutions Calculus III Math Spring 7 In-term exam April th. Suggested solutions This exam contains sixteen problems numbered through 6. Problems 5 are multiple choice problems, which each count 5% of your total

More information

Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint.

Name: Class: Date: 1. Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. . Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f (x, y) = x y, x + y = 8. Set up the triple integral of an arbitrary continuous function

More information

Multivariate Calculus: Review Problems for Examination Two

Multivariate Calculus: Review Problems for Examination Two Multivariate Calculus: Review Problems for Examination Two Note: Exam Two is on Tuesday, August 16. The coverage is multivariate differential calculus and double integration. You should review the double

More information

38. Triple Integration over Rectangular Regions

38. Triple Integration over Rectangular Regions 8. Triple Integration over Rectangular Regions A rectangular solid region S in R can be defined by three compound inequalities, a 1 x a, b 1 y b, c 1 z c, where a 1, a, b 1, b, c 1 and c are constants.

More information

MA 174: Multivariable Calculus Final EXAM (practice) NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper.

MA 174: Multivariable Calculus Final EXAM (practice) NO CALCULATORS, BOOKS, OR PAPERS ARE ALLOWED. Use the back of the test pages for scrap paper. MA 174: Multivariable alculus Final EXAM (practice) NAME lass Meeting Time: NO ALULATOR, BOOK, OR PAPER ARE ALLOWED. Use the back of the test pages for scrap paper. Points awarded 1. (5 pts). (5 pts).

More information

Math Exam III Review

Math Exam III Review Math 213 - Exam III Review Peter A. Perry University of Kentucky April 10, 2019 Homework Exam III is tonight at 5 PM Exam III will cover 15.1 15.3, 15.6 15.9, 16.1 16.2, and identifying conservative vector

More information

MATH 234. Excercises on Integration in Several Variables. I. Double Integrals

MATH 234. Excercises on Integration in Several Variables. I. Double Integrals MATH 234 Excercises on Integration in everal Variables I. Double Integrals Problem 1. D = {(x, y) : y x 1, 0 y 1}. Compute D ex3 da. Problem 2. Find the volume of the solid bounded above by the plane 3x

More information

Multivariate Calculus Review Problems for Examination Two

Multivariate Calculus Review Problems for Examination Two Multivariate Calculus Review Problems for Examination Two Note: Exam Two is on Thursday, February 28, class time. The coverage is multivariate differential calculus and double integration: sections 13.3,

More information

Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates

Integration using Transformations in Polar, Cylindrical, and Spherical Coordinates ections 15.4 Integration using Transformations in Polar, Cylindrical, and pherical Coordinates Cylindrical Coordinates pherical Coordinates MATH 127 (ection 15.5) Applications of Multiple Integrals The

More information

Parametric Surfaces. Substitution

Parametric Surfaces. Substitution Calculus Lia Vas Parametric Surfaces. Substitution Recall that a curve in space is given by parametric equations as a function of single parameter t x = x(t) y = y(t) z = z(t). A curve is a one-dimensional

More information

) in the k-th subbox. The mass of the k-th subbox is M k δ(x k, y k, z k ) V k. Thus,

) in the k-th subbox. The mass of the k-th subbox is M k δ(x k, y k, z k ) V k. Thus, 1 Triple Integrals Mass problem. Find the mass M of a solid whose density (the mass per unit volume) is a continuous nonnegative function δ(x, y, z). 1. Divide the box enclosing into subboxes, and exclude

More information

Chapter 15 Vector Calculus

Chapter 15 Vector Calculus Chapter 15 Vector Calculus 151 Vector Fields 152 Line Integrals 153 Fundamental Theorem and Independence of Path 153 Conservative Fields and Potential Functions 154 Green s Theorem 155 urface Integrals

More information

Math 113 Calculus III Final Exam Practice Problems Spring 2003

Math 113 Calculus III Final Exam Practice Problems Spring 2003 Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross

More information

MATH 2023 Multivariable Calculus

MATH 2023 Multivariable Calculus MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set

More information

MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals)

MA 243 Calculus III Fall Assignment 1. Reading assignments are found in James Stewart s Calculus (Early Transcendentals) MA 43 Calculus III Fall 8 Dr. E. Jacobs Assignments Reading assignments are found in James Stewart s Calculus (Early Transcendentals) Assignment. Spheres and Other Surfaces Read. -. and.6 Section./Problems

More information

Calculus IV. Exam 2 November 13, 2003

Calculus IV. Exam 2 November 13, 2003 Name: Section: Calculus IV Math 1 Fall Professor Ben Richert Exam November 1, Please do all your work in this booklet and show all the steps. Calculators and note-cards are not allowed. Problem Possible

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Calculus III-Final review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the corresponding position vector. 1) Define the points P = (-,

More information

QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 252 FALL 2008 KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY % POSSIBLE

QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 252 FALL 2008 KUNIYUKI SCORED OUT OF 125 POINTS MULTIPLIED BY % POSSIBLE QUIZ 4 (CHAPTER 17) SOLUTIONS MATH 5 FALL 8 KUNIYUKI SCORED OUT OF 15 POINTS MULTIPLIED BY.84 15% POSSIBLE 1) Reverse the order of integration, and evaluate the resulting double integral: 16 y dx dy. Give

More information

MATH 116 REVIEW PROBLEMS for the FINAL EXAM

MATH 116 REVIEW PROBLEMS for the FINAL EXAM MATH 116 REVIEW PROBLEMS for the FINAL EXAM The following questions are taken from old final exams of various calculus courses taught in Bilkent University 1. onsider the line integral (2xy 2 z + y)dx

More information

18.02 Final Exam. y = 0

18.02 Final Exam. y = 0 No books, notes or calculators. 5 problems, 50 points. 8.0 Final Exam Useful formula: cos (θ) = ( + cos(θ)) Problem. (0 points) a) (5 pts.) Find the equation in the form Ax + By + z = D of the plane P

More information

MAC2313 Test 3 A E g(x, y, z) dy dx dz

MAC2313 Test 3 A E g(x, y, z) dy dx dz MAC2313 Test 3 A (5 pts) 1. If the function g(x, y, z) is integrated over the cylindrical solid bounded by x 2 + y 2 = 3, z = 1, and z = 7, the correct integral in Cartesian coordinates is given by: A.

More information

Math 241, Final Exam. 12/11/12.

Math 241, Final Exam. 12/11/12. Math, Final Exam. //. No notes, calculator, or text. There are points total. Partial credit may be given. ircle or otherwise clearly identify your final answer. Name:. (5 points): Equation of a line. Find

More information

Updated: March 31, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University

Updated: March 31, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University Updated: March 3, 26 Calculus III Section 5.6 Math 232 Calculus III Brian Veitch Fall 25 Northern Illinois University 5.6 Triple Integrals In order to build up to a triple integral let s start back at

More information

Math 6A Practice Problems III

Math 6A Practice Problems III Math 6A Practice Problems III Written by Victoria Kala vtkala@math.ucsb.edu H 63u Office Hours: R 1:3 1:3pm Last updated 6//16 Answers 1. 3. 171 1 3. π. 5. a) 8π b) 8π 6. 7. 9 3π 3 1 etailed olutions 1.

More information

1 Double Integrals over Rectangular Regions

1 Double Integrals over Rectangular Regions Contents ouble Integrals over Rectangular Regions ouble Integrals Over General Regions 7. Introduction.................................... 7. Areas of General Regions............................. 9.3 Region

More information

10.7 Triple Integrals. The Divergence Theorem of Gauss

10.7 Triple Integrals. The Divergence Theorem of Gauss 10.7 riple Integrals. he Divergence heorem of Gauss We begin by recalling the definition of the triple integral f (x, y, z) dv, (1) where is a bounded, solid region in R 3 (for example the solid ball {(x,

More information

UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 5

UNIVERSITI TEKNOLOGI MALAYSIA SSE 1893 ENGINEERING MATHEMATICS TUTORIAL 5 UNIVERSITI TEKNOLOGI MALAYSIA SSE 189 ENGINEERING MATHEMATIS TUTORIAL 5 1. Evaluate the following surface integrals (i) (x + y) ds, : part of the surface 2x+y+z = 6 in the first octant. (ii) (iii) (iv)

More information

Multiple Integrals. max x i 0

Multiple Integrals. max x i 0 Multiple Integrals 1 Double Integrals Definite integrals appear when one solves Area problem. Find the area A of the region bounded above by the curve y = f(x), below by the x-axis, and on the sides by

More information

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Summary Assignments...2

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Summary Assignments...2 PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Summary...1 3. Assignments...2 i PMTH212, Multivariable Calculus Assignment Summary 2010 Assignment Date to be Posted

More information

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V

Worksheet 3.4: Triple Integrals in Cylindrical Coordinates. Warm-Up: Cylindrical Volume Element d V Boise State Math 275 (Ultman) Worksheet 3.4: Triple Integrals in Cylindrical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find

More information

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH GOALS Be able to set up and evaluate triple integrals using rectangular, cylindrical, and spherical coordinates MATH TRIPLE INTEGRALS We integrate functions

More information

MIDTERM. Section: Signature:

MIDTERM. Section: Signature: MIDTERM Math 32B 8/8/2 Name: Section: Signature: Read all of the following information before starting the exam: Check your exam to make sure all pages are present. NO CALCULATORS! Show all work, clearly

More information

Contents. 3 Multiple Integration. 3.1 Double Integrals in Rectangular Coordinates

Contents. 3 Multiple Integration. 3.1 Double Integrals in Rectangular Coordinates Calculus III (part 3): Multiple Integration (by Evan Dummit, 8, v. 3.) Contents 3 Multiple Integration 3. Double Integrals in Rectangular Coordinates............................... 3.. Double Integrals

More information

MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU

MATH203 Calculus. Dr. Bandar Al-Mohsin. School of Mathematics, KSU School of Mathematics, KSU Theorem The rectangular coordinates (x, y, z) and the cylindrical coordinates (r, θ, z) of a point P are related as follows: x = r cos θ, y = r sin θ, tan θ = y x, r 2 = x 2

More information

Math 209 (Fall 2007) Calculus III. Solution #5. 1. Find the minimum and maximum values of the following functions f under the given constraints:

Math 209 (Fall 2007) Calculus III. Solution #5. 1. Find the minimum and maximum values of the following functions f under the given constraints: Math 9 (Fall 7) Calculus III Solution #5. Find the minimum and maximum values of the following functions f under the given constraints: (a) f(x, y) 4x + 6y, x + y ; (b) f(x, y) x y, x + y 6. Solution:

More information

First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters.

First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). Surfaces will need two parameters. Math 55 - Vector Calculus II Notes 14.6 urface Integrals Let s develop some surface integrals. First we consider how to parameterize a surface (similar to a parameterized curve for line integrals). urfaces

More information

Solution of final examination

Solution of final examination of final examination Math 20, pring 201 December 9, 201 Problem 1 Let v(t) (2t e t ) i j + π cos(πt) k be the velocity of a particle with initial position r(0) ( 1, 0, 2). Find the accelaration at the

More information

1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two:

1. Suppose that the equation F (x, y, z) = 0 implicitly defines each of the three variables x, y, and z as functions of the other two: Final Solutions. Suppose that the equation F (x, y, z) implicitly defines each of the three variables x, y, and z as functions of the other two: z f(x, y), y g(x, z), x h(y, z). If F is differentiable

More information

1 Double Integral. 1.1 Double Integral over Rectangular Domain

1 Double Integral. 1.1 Double Integral over Rectangular Domain Double Integral. Double Integral over Rectangular Domain As the definite integral of a positive function of one variable represents the area of the region between the graph and the x-asis, the double integral

More information

Math 241, Exam 3 Information.

Math 241, Exam 3 Information. Math 241, xam 3 Information. 11/28/12, LC 310, 11:15-12:05. xam 3 will be based on: Sections 15.2-15.4, 15.6-15.8. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z.

6. Find the equation of the plane that passes through the point (-1,2,1) and contains the line x = y = z. Week 1 Worksheet Sections from Thomas 13 th edition: 12.4, 12.5, 12.6, 13.1 1. A plane is a set of points that satisfies an equation of the form c 1 x + c 2 y + c 3 z = c 4. (a) Find any three distinct

More information

Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007

Math 2374 Spring 2007 Midterm 3 Solutions - Page 1 of 6 April 25, 2007 Math 374 Spring 7 Midterm 3 Solutions - Page of 6 April 5, 7. (3 points) Consider the surface parametrized by (x, y, z) Φ(x, y) (x, y,4 (x +y )) between the planes z and z 3. (i) (5 points) Set up the

More information

f(x, y, z)dv = As remarked above, triple integrals can be evaluated as iterated integrals.

f(x, y, z)dv = As remarked above, triple integrals can be evaluated as iterated integrals. 7.5 Triple Integrals These are just like double integrals, but with another integration to perform. Although this is conceptually a simple extension of the idea, in practice it can get very complicated.

More information

Dr. Allen Back. Nov. 21, 2014

Dr. Allen Back. Nov. 21, 2014 Dr. Allen Back of Nov. 21, 2014 The most important thing you should know (e.g. for exams and homework) is how to setup (and perhaps compute if not too hard) surface integrals, triple integrals, etc. But

More information

MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS

MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT203 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT203 covers essentially the same material as MAT201, but is more in depth and theoretical. Exam problems are often more sophisticated in scope and difficulty

More information

Name: Final Exam Review. (b) Reparameterize r(t) with respect to arc length measured for the point (1, 0, 1) in the direction of increasing t.

Name: Final Exam Review. (b) Reparameterize r(t) with respect to arc length measured for the point (1, 0, 1) in the direction of increasing t. MATH 127 ALULU III Name: 1. Let r(t) = e t i + e t sin t j + e t cos t k (a) Find r (t) Final Exam Review (b) Reparameterize r(t) with respect to arc length measured for the point (1,, 1) in the direction

More information

MATH 251 Fall 2016 EXAM III - VERSION A

MATH 251 Fall 2016 EXAM III - VERSION A MATH 51 Fall 16 EXAM III - VERSION A LAST NAME: FIRST NAME: SECTION NUMBER: UIN: DIRECTIONS: 1. You may use a calculator on this exam.. TURN OFF cell phones and put them away. If a cell phone is seen during

More information

Math 253, Section 102, Fall 2006 Practice Final Solutions

Math 253, Section 102, Fall 2006 Practice Final Solutions Math 253, Section 102, Fall 2006 Practice Final Solutions 1 2 1. Determine whether the two lines L 1 and L 2 described below intersect. If yes, find the point of intersection. If not, say whether they

More information

A small review, Second Midterm, Calculus 3, Prof. Montero 3450: , Fall 2008

A small review, Second Midterm, Calculus 3, Prof. Montero 3450: , Fall 2008 A small review, Second Midterm, Calculus, Prof. Montero 45:-4, Fall 8 Maxima and minima Let us recall first, that for a function f(x, y), the gradient is the vector ( f)(x, y) = ( ) f f (x, y); (x, y).

More information

Dr. Allen Back. Nov. 19, 2014

Dr. Allen Back. Nov. 19, 2014 Why of Dr. Allen Back Nov. 19, 2014 Graph Picture of T u, T v for a Lat/Long Param. of the Sphere. Why of Graph Basic Picture Why of Graph Why Φ(u, v) = (x(u, v), y(u, v), z(u, v)) Tangents T u = (x u,

More information

MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points.

MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. MATH. 2153, Spring 16, MWF 12:40 p.m. QUIZ 1 January 25, 2016 PRINT NAME A. Derdzinski Show all work. No calculators. The problem is worth 10 points. 1. Evaluate the area A of the triangle with the vertices

More information

Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures

Grad operator, triple and line integrals. Notice: this material must not be used as a substitute for attending the lectures Grad operator, triple and line integrals Notice: this material must not be used as a substitute for attending the lectures 1 .1 The grad operator Let f(x 1, x,..., x n ) be a function of the n variables

More information

R f da (where da denotes the differential of area dxdy (or dydx)

R f da (where da denotes the differential of area dxdy (or dydx) Math 28H Topics for the second exam (Technically, everything covered on the first exam, plus) Constrained Optimization: Lagrange Multipliers Most optimization problems that arise naturally are not unconstrained;

More information

There are 10 problems, with a total of 150 points possible. (a) Find the tangent plane to the surface S at the point ( 2, 1, 2).

There are 10 problems, with a total of 150 points possible. (a) Find the tangent plane to the surface S at the point ( 2, 1, 2). Instructions Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper. You may use a scientific

More information

Final Exam Review. Name: Class: Date: Short Answer

Final Exam Review. Name: Class: Date: Short Answer Name: Class: Date: ID: A Final Exam Review Short Answer 1. Find the distance between the sphere (x 1) + (y + 1) + z = 1 4 and the sphere (x 3) + (y + ) + (z + ) = 1. Find, a a + b, a b, a, and 3a + 4b

More information

Background for Surface Integration

Background for Surface Integration Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to

More information

Integration. Example Find x 3 dx.

Integration. Example Find x 3 dx. Integration A function F is called an antiderivative of the function f if F (x)=f(x). The set of all antiderivatives of f is called the indefinite integral of f with respect to x and is denoted by f(x)dx.

More information

Math 32B Discussion Session Week 2 Notes January 17 and 24, 2017

Math 32B Discussion Session Week 2 Notes January 17 and 24, 2017 Math 3B Discussion Session Week Notes January 7 and 4, 7 This week we ll finish discussing the double integral for non-rectangular regions (see the last few pages of the week notes) and then we ll touch

More information

University of California, Berkeley

University of California, Berkeley University of California, Berkeley FINAL EXAMINATION, Fall 2012 DURATION: 3 hours Department of Mathematics MATH 53 Multivariable Calculus Examiner: Sean Fitzpatrick Total: 100 points Family Name: Given

More information

8(x 2) + 21(y 1) + 6(z 3) = 0 8x + 21y + 6z = 55.

8(x 2) + 21(y 1) + 6(z 3) = 0 8x + 21y + 6z = 55. MATH 24 -Review for Final Exam. Let f(x, y, z) x 2 yz + y 3 z x 2 + z, and a (2,, 3). Note: f (2xyz 2x, x 2 z + 3y 2 z, x 2 y + y 3 + ) f(a) (8, 2, 6) (a) Find all stationary points (if any) of f. et f.

More information

Outcomes List for Math Multivariable Calculus (9 th edition of text) Spring

Outcomes List for Math Multivariable Calculus (9 th edition of text) Spring Outcomes List for Math 200-200935 Multivariable Calculus (9 th edition of text) Spring 2009-2010 The purpose of the Outcomes List is to give you a concrete summary of the material you should know, and

More information

Homework 8. Due: Tuesday, March 31st, 2009

Homework 8. Due: Tuesday, March 31st, 2009 MATH 55 Applied Honors Calculus III Winter 9 Homework 8 Due: Tuesday, March 3st, 9 Section 6.5, pg. 54: 7, 3. Section 6.6, pg. 58:, 3. Section 6.7, pg. 66: 3, 5, 47. Section 6.8, pg. 73: 33, 38. Section

More information

The diagram above shows a sketch of the curve C with parametric equations

The diagram above shows a sketch of the curve C with parametric equations 1. The diagram above shows a sketch of the curve C with parametric equations x = 5t 4, y = t(9 t ) The curve C cuts the x-axis at the points A and B. (a) Find the x-coordinate at the point A and the x-coordinate

More information

Math 32B Discussion Session Week 2 Notes April 5 and 7, 2016

Math 32B Discussion Session Week 2 Notes April 5 and 7, 2016 Math 3B Discussion Session Week Notes April 5 and 7, 6 We have a little flexibility this week: we can tie up some loose ends from double integrals over vertically or horizontally simple regions, we can

More information

Chapter 15: Functions of Several Variables

Chapter 15: Functions of Several Variables Chapter 15: Functions of Several Variables Section 15.1 Elementary Examples a. Notation: Two Variables b. Example c. Notation: Three Variables d. Functions of Several Variables e. Examples from the Sciences

More information

Ma MULTIPLE INTEGRATION

Ma MULTIPLE INTEGRATION Ma 7 - MULTIPLE INTEGATION emark: The concept of a function of one variable in which y gx may be extended to two or more variables. If z is uniquely determined by values of the variables x and y, thenwesayz

More information

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Introduction Timetable Assignments...

PURE MATHEMATICS 212 Multivariable Calculus CONTENTS. Page. 1. Assignment Summary... i 2. Introduction Timetable Assignments... PURE MATHEMATICS 212 Multivariable Calculus CONTENTS Page 1. Assignment Summary... i 2. Introduction...1 3. Timetable... 3 4. Assignments...5 i PMTH212, Multivariable Calculus Assignment Summary 2009

More information

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv.

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv. MAC2313 Final A (5 pts) 1. Let f(x, y, z) be a function continuous in R 3 and let S be a surface parameterized by r(u, v) with the domain of the parameterization given by R; how many of the following are

More information

Vectors and the Geometry of Space

Vectors and the Geometry of Space Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c

More information

f xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is

f xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is 1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,

More information

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = enter NONE.

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = enter NONE. Ch17 Practice Test Sketch the vector field F. F(x, y) = (x - y)i + xj Evaluate the line integral, where C is the given curve. C xy 4 ds. C is the right half of the circle x 2 + y 2 = 4 oriented counterclockwise.

More information

MA FINAL EXAM Green April 30, 2018 EXAM POLICIES

MA FINAL EXAM Green April 30, 2018 EXAM POLICIES MA 6100 FINAL EXAM Green April 0, 018 NAME STUDENT ID # YOUR TA S NAME RECITATION TIME Be sure the paper you are looking at right now is GREEN! Write the following in the TEST/QUIZ NUMBER boxes (and blacken

More information

Coordinate Transformations in Advanced Calculus

Coordinate Transformations in Advanced Calculus Coordinate Transformations in Advanced Calculus by Sacha Nandlall T.A. for MATH 264, McGill University Email: sacha.nandlall@mail.mcgill.ca Website: http://www.resanova.com/teaching/calculus/ Fall 2006,

More information

1.(6pts) Which integral computes the area of the quarter-disc of radius a centered at the origin in the first quadrant? rdr d

1.(6pts) Which integral computes the area of the quarter-disc of radius a centered at the origin in the first quadrant? rdr d .(6pts) Which integral computes the area of the quarter-disc of radius a centered at the origin in the first quadrant? (a) / Z a rdr d (b) / Z a rdr d (c) Z a dr d (d) / Z a dr d (e) / Z a a rdr d.(6pts)

More information

MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM

MATH 2400: CALCULUS 3 MAY 9, 2007 FINAL EXAM MATH 4: CALCULUS 3 MAY 9, 7 FINAL EXAM I have neither given nor received aid on this exam. Name: 1 E. Kim................ (9am) E. Angel.............(1am) 3 I. Mishev............ (11am) 4 M. Daniel...........

More information

Topic 6: Calculus Integration Volume of Revolution Paper 2

Topic 6: Calculus Integration Volume of Revolution Paper 2 Topic 6: Calculus Integration Standard Level 6.1 Volume of Revolution Paper 1. Let f(x) = x ln(4 x ), for < x

More information

To find the maximum and minimum values of f(x, y, z) subject to the constraints

To find the maximum and minimum values of f(x, y, z) subject to the constraints Midterm 3 review Math 265 Fall 2007 14.8. Lagrange Multipliers. Case 1: One constraint. To find the maximum and minimum values of f(x, y, z) subject to the constraint g(x, y, z) = k: Step 1: Find all values

More information

Exam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA:

Exam 3 SCORE. MA 114 Exam 3 Spring Section and/or TA: MA 114 Exam 3 Spring 217 Exam 3 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test.

More information

f x = 2e xy +y(2x+y)e xy = (2+2xy+y 2 )e xy.

f x = 2e xy +y(2x+y)e xy = (2+2xy+y 2 )e xy. gri (rg38778) Homework 11 gri (11111) 1 This print-out should have 3 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Find lim (x,y) (,) 1

More information

Chapter 5 Partial Differentiation

Chapter 5 Partial Differentiation Chapter 5 Partial Differentiation For functions of one variable, y = f (x), the rate of change of the dependent variable can dy be found unambiguously by differentiation: f x. In this chapter we explore

More information

University of Saskatchewan Department of Mathematics & Statistics MATH Final Instructors: (01) P. J. Browne (03) B. Friberg (05) H.

University of Saskatchewan Department of Mathematics & Statistics MATH Final Instructors: (01) P. J. Browne (03) B. Friberg (05) H. University of Saskatchewan Department of Mathematics & Statistics MATH. Final Instructors: (0) P. J. Browne (0) B. Friberg (0) H. Teismann December 9, 000 Time: :00-:00 pm This is an open book exam. Students

More information

Worksheet 3.5: Triple Integrals in Spherical Coordinates. Warm-Up: Spherical Coordinates (ρ, φ, θ)

Worksheet 3.5: Triple Integrals in Spherical Coordinates. Warm-Up: Spherical Coordinates (ρ, φ, θ) Boise State Math 275 (Ultman) Worksheet 3.5: Triple Integrals in Spherical Coordinates From the Toolbox (what you need from previous classes) Know what the volume element dv represents. Be able to find

More information

1 Vector Functions and Space Curves

1 Vector Functions and Space Curves ontents 1 Vector Functions and pace urves 2 1.1 Limits, Derivatives, and Integrals of Vector Functions...................... 2 1.2 Arc Length and urvature..................................... 2 1.3 Motion

More information