Exam III Sample Questions
1. Set up an integral to find the volume that is inside the sphere
x2 + y 2 + z 2 = c2 and outside the cylinder x2 + y 2 = b2 where b < c.
Evaluate this integral. If the cylindrical hole has height h above the xyplane write the equation for the volume in terms of h.
2. Let C be a path along x = y 2 − y 3 from (2, −1) to (−4, 2).
´
(a) Set up but do not evaluate C xy + y 2 ds.
´
(b) Evaluate C xy dx + y 2 dy.
(c) Evaluate
´
C
F · dr
with F(x, y) = hxy, y 2 i
3. An ice-cream cone is in the shape of the solid bounded
above by the sphere
p
x2 + y 2 + z 2 = 9 and below by the cone z = 3x2 + 3y 2 with constant
density K.
(a) Find the mass of the ice-cream cone.
(b) Find the z-coordinate of the center of mass of the ice cream cone.
4. A particle moves under a force field F(x, y, z) = zi + yj + xk. Find
the work done by the force in moving an object along the twisted cubic
r(t) = ht3 , t2 , ti.
´
5. Evaluate the line integral C −x2 ydx + xy 2 dy where C is the path that
starts from (0, 0) and travels along the x-axis to (−2, 0), then
√ travels
√
counterclockwise around the circle of radius 2 to the point ( 2, − 2),
from there along the line y = −x back to (0, 0).
6. Let F = he−y + zex , −xe−y − ez , −yez + ex i.
(a) Show this vector field is conservative.
~ = F.
(b) Find f (x, y, z) so ∇f
(c) Evaluate the line integral
h0, 1, 2i
´
C
F · dr if C is the line from h1, 0, 0i to
7. Do you think the vector field depicted in the following graph is conservative
or not? Make drawings on the vector field as needed and explain your
answer.
8. Find the area of the cap of the sphere x2 + y 2 + z 2 = 4 which lies above
the plane z = 1.
9. Find an equation to the plane tangent to the surface parametrized by
x = u + v, y = 3u2 , z = u − v at the point (2, 3, 0).
¨
10. Evaluate
xy dS, where S is the lateral part of the cylinder x2 + z 2 = 1
S
that lies between the planes y = 0 and x + y = 2.
11. Verify Stokes’ Theorem if S is the paraboloid z = 4 − x2 − y 2 which lies
above the x-y plane oriented upward and F(x, y, z) = yi + zj + xk.