Double Integrals over Polar Coordinate

Transcription

1 DOUBLE INTEGRALS OVER POLAR COORDINATE Double Integrals over Polar Coordinate 1. Polar Coordinates. The polar coordinates (r, θ) of a point are related to the rectangular coordinates (x,y) by the equations r 2 = x 2 + y 2 x = r cosθ y = r sinθ (1) The regions in the Figure is a special case of polar rectangle: R = {(r,θ) a r b,α θ β} (2) If f is continuous on a polar rectangle R given by a r b, α θ β, where β α 2π, then R f (x, y)d A = β b α a f (r cosθ,r sinθ)r dr dθ (3) The formula says that we convert from rectangular to polar coordinates in a double integral by writing x = r cosθ and y = r sinθ, using the appropriate limits of integration for r and θ, and replacing d A by r dr dθ. Exercise 1. Evaluate R (3x + 4y 2 )d A, where R is the region in the upper half-plane bounded by the circles x 2 + y 2 = 1 and x 2 + y 2 = 4 1

2 DOUBLE INTEGRALS OVER POLAR COORDINATE Sol: R (3x + 4y 2 )d A = π 2 1 (3r cosθ + 4(r sinθ) 2 )r dr dθ Exercise 2. Find the volume of the solid bounded by the plane z = and the paraboloid z = 1 x 2 y 2. Sol: f (x, y) = 1 x 2 y 2, r 2 = x 2 + y 2 1, r 1 and θ 2π. 2π 1 (1 r 2 )r dr dθ 2. If f is continuous on a polar region D given by D = {(r,θ) α θ β,h 1 (θ) r h 2 (θ)} then R f (x, y)d A = β h2 (θ) α h 1 (θ) f (r cosθ,r sinθ)r dr dθ (4) 2

3 DOUBLE INTEGRALS OVER POLAR COORDINATE Exercise 3. Find the area enclosed by one loop of the four leaved rose r = cos2θ. Sol: D = {(r,θ) π 4 θ π, r cos2θ}. Then the area is 4 π 4 A = π 4 cos2θ r dr dθ (5) Exercise 4. Find the volume of the solid that lies under the paraboloid z = x 2 + y 2 above the xy-plane, and inside the cylinder x 2 + y 2 = 2x. The solid lies above the disk D whose boundary circle has equation x 2 + y 2 = 2x which is 3

4 DOUBLE INTEGRALS OVER POLAR COORDINATE same as (x 1) 2 + y 2 = 1 The polar equation of circle with radius R and centered at (R,) is r = 2R cosθ. Then D = {(r,θ) π 2 θ π 2, r 2cosθ}. And f (x, y) = x2 + y 2 = r 2. π 2 V = π 2 2cosθ r 2 r dr dθ 4

5 PROBLEM SET Problem Set 1. Evaluate the given integral by polar coordinates (a) D x2 yd A where D is the top half of the disk with center the origin and radius 5. (b) D (2x y)d A where D is the region in the first quadrant enclosed by the circle x 2 + y 2 = 4 and the lines x = and y = x (c) D sin(x2 + y 2 )d A where D is the region in the first quadrant between the circles with center origin and radii 1 and 3. (d) y 2 D d A where D is the region lies between the circles x 2 + y 2 = a 2 and x 2 + y 2 = x 2 +y 2 b 2 with < a < b. 2. Use double integral to find the area. (a) One loop of the rose r = cos(3θ) (b) Region enclosed by r = 1 + cosθ and r = 1 cosθ (c) Region inside the circle (x 1) 2 + y 2 = 1 and outside the circle x 2 + y 2 = 1 3. Find volume of the given solid (a) Under z = x 2 + y 2 and above x 2 + y 2 4 (b) Below z = 18 2x 2 2y 2 and above xy-plane (c) Inside x 2 + y 2 + z 2 = 16 and outside x 2 + y 2 = Convert the iterated integral into integral with polar coordinates: (a) 3 9 x 2 3 sin(x 2 + y 2 )d yd x (b) 1 2 y 2 y (x + y)d yd x 5

6 TRIPLE INTEGRAL Triple Integral 1. Review. (a) Function of one variable. y = f (x) when a x b. Divide [a,b] into n equal length sub-intervals [x, x 1 ], [x 1, x 2 ],...,[x n 1, x n ]. The length of each sub-intervals x = b a n. Sample points from each sub-intervals are x1, x 2,...,x n. The Riemann sum is R n = n i=1 f (x i ) x. b a f (x)d x R n b a f (x)d x = lim n R n (b) Function of two variable. z = f (x, y) when a x b and c y d. Divide [a,b] into m equal length sub-intervals [x, x 1 ], [x 1, x 2 ],...,[x m 1, x m ]. Divide [c,d] into n equal length sub-intervals [y, y 1 ], [y 1, y 2 ],...,[y n 1, y n ]. Then x = b a d c and y = m n. The area of each sub-rectangles A = x y. The Riemann double sum is R mn = m n i=1 j =1 f (x i, y j ) A. R f (x, y)d A R mn R f (x, y)d A = lim m,n R mn 2. Function of three variable. w = f (x, y, z) when a x b, c y d and r z s. Divide [a,b] into l equal length sub-intervals [x, x 1 ], [x 1, x 2 ],...,[x l 1, x l ]. Divide [c,d] into m equal length sub-intervals [y, y 1 ], [y 1, y 2 ],...,[y m 1, y m ]. Divide [c,d] into n equal length sub-intervals [z, z 1 ], [z 1, z 2 ],...,[z n 1, z n ]. Then x = b a, y = d c l m s r and z = n. The volume of each sub-boxes V = x y z. The Riemann triple sum is R l mn = l i=1 m j =1 n k=1 f (x i, y j, z k ) V. 6

7 TRIPLE INTEGRAL R f (x, y, z)dv R lmn R f (x, y, z)dv = lim m,n,l R lmn. 3. Fubini s Theorem for Triple Integral. If f is continuous on the rectangular box B = [a, b] [c,d] [r, s], then B f (x, y, z)dv = s d b r c a f (x, y, z)d xd yd z Exercise 5. Evaluate the triple integral B x y z2 dv where B = [,1] [ 1,2] [,3] sol: You can choose any orders of integration. For example you can choose to do integration with respect to x, y and then z. B x y z 2 dv = Or you can do integration with respect to y, z, and then x. B x y z 2 dv = x y z 2 d xd yd z x y z 2 d yd zd x 4. Type I Region. A region E is type I if E = {(x, y, z) (x, y) D,u 1 (x, y) z u 2 (x, y)} (6) Then B u2 (x,y) f (x, y, z)dv = [ f (x, y, z)d z]d A D u 1 (x,y) There are two situations for type I. (a) If E = {(x, y, z) a x b, g 1 (x) y g 2 (x),u 1 (x, y) z u 2 (x, y)} (7) then B f (x, y, z)dv = b g2 (x) u2 (x,y) a g 1 (x) [ u 1 (x,y) f (x, y, z)d z]d yd x (b) If E = {(x, y, z) c y d,h 1 (y) x h 2 (y),u 1 (x, y) z u 2 (x, y)} (8) 7

8 TRIPLE INTEGRAL then B f (x, y, z)dv = d h2 (y) u2 (x,y) c h 1 (y) [ u 1 (x,y) f (x, y, z)d z]d xd y Exercise 6. Evaluate E zdv where E is the solid bounded by the four planes x =, y =, z = and x + y + z = 1. Sol. E = {(x, y, z) x 1, y 1 x, z 1 x y} Then the volume is E zdv = 1 1 x 1 x y zd zd yd x 5. Type II Region. A region E is type II if E = {(x, y, z) (y, z) D,u 1 (y, z) x u 2 (y, z)} (9) Then E u2 (y,z) f (x, y, z)dv = [ f (x, y, z)d x]d A D u 1 (y,z) 6. Type III Region. A region E is type III if E = {(x, y, z) (x, z) D,u 1 (x, z) y u 2 (x, z)} (1) Then E u2 (x,z) f (x, y, z)dv = [ f (x, y, z)d y]d A D u 1 (x,z) Exercise 7. Evaluate E x 2 + z 2 dv where E is the solid bounded by y = x 2 + z 2 and y = 4. Sol. 8

9 TRIPLE INTEGRAL E = {(x, y, z) 2 x 2, x 2 y 4, y x 2 z y x 2 } Then the volume is E 2 x 2 + z 2 dv = 2 x 2 y 4 x 2 y x 2 y x 2 x 2 + z 2 d zd yd x Exercise 8. (a) Express 1 f (x, y, z)d zd yd x as a triple integral. (b) Then rewrite it as an iterated integral in a different order, integrating first with respect to x, then z and then y. Sol. x 2 y (a) 1 f (x, y, z)d zd yd x = E f (x, y, z)dv where E = {(x, y, z) x 1, y x 2, z y}. (b) If we integrate first with respect to x then z and then y, then E = {(x, y, z) a y b, g 1 (y) z g 2 (y),u 1 (y, z) x u 2 (y, z)}. We need to find a, b, g 1, g 2,u 1 and u 2. To find a and b, use x 1 and y x 2. Then y x = 1. So a = and b = 1. To find g 1 and g 2, use z y. So g 1 = and g 2 = y. To find u 1 and u 2, use x 1 and y x 2. y x 2 and x 1 y x 1. So u 1 = y and u 2 = 1. 9

10 TRIPLE INTEGRAL The new iterated integral is 1 y 1 y f (x, y, z)d xd zd y Exercise 9. Find the volume of the tetrahedron T bounded by the planes x + 2y + z = 2. x = 2y, x = and z =. Sol. V = 1 1 x/2 2 x 2y x/2 d zd yd x 1

11 TRIPLE INTEGRAL 15.7 Problem Set 1. Evaluate E (xz y 3 )dv where E = [ 1,1] [,2] [,1]. 2. Evaluate 2 3. Evaluate 1 z 2 y z 2x x (2x y)d xd yd z y (2x y z)d zd yd x 4. Express the integral E f (x, y, z)dv as an iterated integral in six different ways, where E is the solid bounded by the given surface. (a) y = 4 x 2 4z 2 and y = (b) y 2 + z 2 = 9, x = 2 and x = 2 (c) y = x 2, z =, and y + 2z = 4 (d) x = 2, y = 2, z = and x + y 2z = 2 5. The figure shows the region of integration for the integral 2 1 x 1 y f (x, y, z)d zd yd x Rewrite this integral as an equivalent iterated integral in the five other orders. 6. The figure shows the region of integration for the integral 1 1 x 2 1 x f (x, y, z)d yd zd x Rewrite this integral as an equivalent iterated integral in the five other orders. 11

12 TRIPLE INTEGRAL 12

13 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES Triple Integrals in Cylindrical Coordinates 1. Cylindrical Coordinates. To convert from cylindrical to rectangular coordinates, we use the equation x = r cosθ y = r sinθ z = z To convert from rectangular to cylindrical coordinates, we use r 2 = x 2 + y 2 tanθ = y x z = z Exercise 1. (a) Plot the point with cylindrical coordinates (2, 2π,1) and find its rectangular 3 coordinates. (b) Find cylindrical coordinates of the point with rectangular coordinates (3,-3,-7) (c) Describe the surface whose equation in cylindrical coordinates is z = r 13

14 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 2. Evaluating Triple Integrals with Cylindrical Coordinates. If f is continuous on region E and E is a type I region: where D is given in polar coordinates by E = {(x, y, z) (x, y) D,u 1 (x, y) z u 2 (x, y)} (11) D = {(x, y) α θ β,h 1 (θ) r h 2 (θ)} (12) then E f (x, y, z)dv = β h2 (θ) u2 (r cosθ,r sinθ) α h 1 (θ) u 1 (r cosθ,r sinθ) f (r cosθ,r sinθ, z)r d zdr dθ Exercise 11. A solid E lies within the cylinder x 2 + y 2 = 1, below the plane z = 4, and above the paraboloid z = 1 x 2 y 2. (See Figure 8.) The density at any point is proportional to its distance from the axis of the cylinder. Find the mass of E. 14

15 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES Exercise 12. Evaluate 2 4 x x 2 x 2 +y 2(x2 + y 2 )d zd yd x 15

16 TRIPLE INTEGRALS IN CYLINDRICAL COORDINATES 1. Find the rectangular coordinates (a) (4, π 3, 2) (b) (2, π 2,1) (c) (1,1,1) 2. Find the cylindrical coordinates (a) ( 1, 1, 1) (b) (4, 3, 2) 15.8 Problem Set 3. Write the equations in cylindrical coordinates. (a) x 2 x + y 2 + z 2 = 1 (b) z = x 2 y 2 (c) 3x + 2y + z = 6 4. Sketch the solid describted by the given inequalities (a) r leq2, π 2 θ π 2, z 1 (b) θ π 2, r z 2 5. Evaluate E x 2 + y 2 dv where E is the region that lies inside the cylinder x 2 + y 2 = 16 and between the planes z = -5 and z = Evaluate E x + y + zdv where E is the solid in the first octant that lies under the paraboloid z = 16 x 2 y Evaluate E xdv where E is the solid that lines between the cylinders x2 + y 2 = 1 and x 2 + y 2 = 25, above the xy-plane, and below the plane z = y Evaluate the integral, where E is enclosed by the paraboloid z = 4 + x 2 + y 2, the cylinder x 2 + y 2 = 5, and the xy-plane. Use cylindrical coordinates. 9. Find the volume of the solid that is enclosed by the cone z = x 2 + y 2 and the sphere x 2 + y 2 + z 2 = Evaluate the integral by changing to cylindrical coordinates 3 9 x 2 9 x 2 y 2 3 x 2 + y 2 d zd yd x 16

17 TRIPLE INTEGRALS IN SPHERICAL COORDINATES Triple Integrals in Spherical Coordinates 1. Spherical coordinates. The spherical coordinates (ρ,θ,φ) can be converted to rectangular coordinates and x = ρ sinφcosθ y = ρ sinφsinθ z = ρ cosφ ρ 2 = x 2 + y 2 + z 2 (13) Exercise 13. Sketch the point and find its rectangular coordinates. (2, π 4, π 3 ) Exercise 14. Find the spherical coordinates for (,2 3, 2) 2. Evaluating Triple Integrals with Spherical Coordinates. In the spherical coordinate system, the spherical wedge is E = {(ρ,θ,φ), a ρ b,α θ β,c φ d} (14) The formula for triple integration in spherical coordinates is 17

18 TRIPLE INTEGRALS IN SPHERICAL COORDINATES Exercise 15. Evaluate B e(x2 +y 2 +z 2 ) 3/2 dv where B is the unit ball B = {(x, y, z) x 2 + y 2 + z 2 1} Exercise 16. Use spherical coordinates to find the volume of the solid that lies above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = z. 18

19 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 15.9 Problem Set 1. Plot the point and find the rectangular coordinates. (a) (6, π 3, π 6 ) (b) (3, π 2, 3π 4 ) (c) (2, π 2, π 2 ) 2. Change from rectangular to spherical coordinates. (a) (,-2,) (b) ( 1,1, 2) (c) (1,, 3) 3. Describe in words the surface whose equation is given. (a) φ = π 3 (b) ρ = 3 4. Write the equation in spherical coordinates. (a) z 2 = x 2 + y 2 (b) x 2 + y 2 = 9 (c) x 2 2x + y 2 + z 2 = (d) x 2 y 2 z 2 = 1 5. Evaluate the integral. Then sketch the solid whose volume is given by the integral π 6 π 2 5 ρ 2 sin(φ)dρdθdφ 6. Set up the triple integral of an arbitrary continuous function f(x, y, z) in spherical coordinates over the solid shown. 7. Use spherical coordinates to evaluate B (x2 + y 2 +z 2 ) 2 dv where B is the ball with center the origin and radius 5. 19

20 TRIPLE INTEGRALS IN SPHERICAL COORDINATES 8. Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume of the smaller wedge cut from a sphere of radius 8 by two planes that intersect along a diameter at an angle of π Use cylindrical or spherical coordinates, whichever seems more appropriate. 1. Find the volume enclosed by ρ = 8sin(φ) 11. 2