Math 251 Quiz 5 Fall b. Calculate. 2. Sketch the region. Write as one double integral by interchanging the order of integration: 2

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1 Math 251 Quiz 5 Fall a. Calculate x dx dy b. Calculate xdxdy 2. Sketch the region. Write as one double integral by interchanging the order of integration: 0 2 dx 2 x dy f(x,y) dx 2 2x dy f(x,y). 3. Sketch the region D = {(x,y) : 0 y x,0 x 2 + y 2 4}. Calculate (exact expression) x dx dy using polar coordinates. D Approximate your result to two decimals and compare with the area of the (1/8) th -circle to two decimals.

2 Math 251 Quiz 6 Fall 2002 Set up integrals BUT DO NOT EVALUATE 1. Write the formulas for z and r in spherical coordinates. Write a triple integral in spherical coordinates for the integral of f(x,y,z) = x 2 over the cone z = x 2 + y 2, 0 z 8. Set it up to integrate in the order: ρ first, φ, then θ last. 2. Sketch the region R in the xy-plane bounded by the lines y = 0, y = x, x + y = 1, x + y = 4. Indicate/Label intersection points clearly. For the change-of-variables: u = y/x,v = x + y, sketch the corresponding region in the uv-plane and write the integral x 2 da as an iterated integral in the u,v -coordinates. Note that x = v/(1 + u),y = v(1 (1 + u) 1 ) is the inverse transformation. R

3 1. Matching. Fill in the appropriate letter. 1. ds for a surface z = g(x,y) A. r u r v dudv 2. ds for a surface r(u,v) B. r u r v dudv 3. ds for any surface C. G x G z, G y G z,1 4. Unit normal ˆn to the D. g x, g y,1 dxdy surface G(x,y,z) = c 5. one version of ds E. for a surface G(x, y, z) = c G G 6. surface of revolution F. ds G. x = ucos v, y = f(u), z = usin v H. x = u + v, y = v, z = u v I. 1 + g 2 x + g 2 y dudv

4 2 2. Match equivalent expressions. 1. gradφ A. F 2. divf B. F 3. curlf C. 2 φ 4. div gradφ, D. φ same as Laplacian of φ 3. a. Let φ = e (x2 +y 2 +z 2 )/2. curl gradφ = b. Let F = sin y + cos y,x(cos y sin y) + y, 2z. div curlf = 4. Referring to these theorems: Fundamental Theorem for Line Integrals, Green s Theorem, Divergence (Gauss ) Theorem, Stokes Theorem As usual, is used to indicate the (one-less-dimensional) boundary of the curve/surface/volume written as C, S, V respectively. Write an equivalent integral or expression using the appropriate theorem. State which theorem you are using. a. S F dr = b. C gradφ dr, where C joins point P to point Q.

5 MATH 251 Fall 1998 QUIZ 5 Ph. Feinsilver ** DO NOT CALCULATE ANY INTEGRALS ** 1. Write two integrals (corresponding to order of integration) for the volume of the solid formed by the surface z = e xy over the triangular region in the x-y plane with vertices (0,0), (2,4), ( 2,4). 2. Interchange the order of integration (include sketch): y 2 sin(x y)dxdy. 3. Write an integral in cylindrical coordinates for the volume of revolution above the surface z = e r sitting over the disk x 2 + y Write an integral in spherical coordinates for volume of the sphere centered at the origin of radius R.

6 MATH 251 Fall 1998 QUIZ 6 Ph. Feinsilver 1. Let f(x,y) = sin(xy), g(x,y,z) = x 2 2xy y z2. Find f, g. 2. Let F = z + xy,xz,xy, G = e y+z,e x+z,e x+y. Find curlf, divg. 3. Fill in: a. curl f = b. G = 4. (continuing #2) Find φ such that F = gradφ if it exists.

7 MATH 251 Fall 1998 QUIZ 7 Ph. Feinsilver Surface area, surface and flux integrals 1. a. Find the area of the surface of revolution r(u, v) = u, sinu cosv, sinu sinv over the region 0 u π, 0 u 2π. b. Show that y 2 + z 2 = sin 2 x on the surface. What is the axis of revolution? In what interval does x vary? c. Intersect the surface with the x-y plane to recover the original curve (before it s revolved). Sketch the curve in the x-y plane, then indicate the surface. 2. Consider the surface r(u, v) = 2 cosu, 2 sinu, v, 0 u π 2, 0 v 2 a. Show that x 2 + y 2 = 4, which is a circle of what radius? Therefore the surface is quarter of a of radius and height. b. Find the area of the surface by calculating ds. c. Evaluate xy ds. S S 3. Find the flux of F = xî + yĵ + zˆk through the first octant of the sphere x 2 + y 2 + z 2 = 16.

8 MATH 251 Fall 1998 QUIZ 8 Ph. Feinsilver Preliminary Study on Stokes Theorem 1. Consider the triangle with vertices A(0, 0, 0), B(0, 3, 0), C(1, 2, 1). a. Show that a normal to the plane containing ABC is 3, 0, 3 and therefore that an equation of the plane is x z = 0. b. Let F = 2y, 3z, x. Let C = the boundary of the triangle, thus consisting of three line segments: C 1 from point A to point B, C 2 from B to C, and C 3 from C to A. Find F dr. C c. Sketch the projection of ABC onto the x-y-plane. Call this region R. d. Calculate F and F ds, where S is the face S of the triangle, using Theorem to convert to an integral over R. e. Explain your answers in regards to Stokes Theorem.