Math 12. Calculus Plus. Written Homework 8.
Due on Wednesday, 11/13/13.
Turn in this homework in the Math 12 box in Kemeny Hall before 10 am on Wednesday.
1. (Chapter 16.6, #26) Find a parametric representation for the part of the plane z = x + 3
that lies inside the cylinder x2 + y 2 = 1.
2. (Chapter 16.6, #36) Let r(u, v) = hsin u, cos u sin v, sin vi. Find an equation for the tangent
plane to this surface at u = π/6, v = π/6.
p
3. (Chapter 16.6, #42) Find the surface area of the part of the cone z = x2 + y 2 that lies
between the plane y = x and the parabolic cylinder y = x2 .
4. (Chapter 16.6, #64a) Find a parametric representation for the torus obtained by rotating
about the z-axis the circle in the xz-plane with center (b, 0, 0) and radius a < b. (See the
textbook for a picture and a relevant hint.)
5. (Chapter 16.6, #64c) Use the parametric representation from the previous problem to find
the surface area of the torus.
p
6. (Chapter 16.7, #4) Suppose that f (x, y,
z)
=
g(
x2 + y 2 + z 2 ), where g is a function of one
RR
variable such that g(2) = −5. Evaluate S f (x, y, z) dS, where S is the sphere x2 +y 2 +z 2 = 4.
7. (Chapter 16.7, #39) Find the center of mass of the hemisphere x2 + y 2 + z 2 = a2 , z ≥ 0, if
it has constant density.
8. (Chapter 16.7, #29) Let F = hx, 2y,
RR3zi, and let S be the cube with vertices (±1, ±1, ±1)
with positive orientation. Evaluate S F · dS.
9. (Chapter 16.7, #49) Let F be an inverse square vector field (that is, F(r) = cr/|r|3 for some
constant c, where r = hx, y, zi). Show that the flux of F across a sphere S centered at the
origin is independent of the radius of S.