MATH261 EXAM III FALL 2013
NAME:
SI:
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Problem Points
1
20
2
20
3
18
4
17
5
25
Total
100
Score
1. (20pts) Using a triple integral, find the volume of the solid in the first octant of R3
bounded by y = x2 , x = y 2 , and x + y − z = −5. Include a sketch of the projection of
the solid onto the xy-plane.
2. Consider a solid mass enclosed by the surfaces (z − 2)2 = x2 + y 2 and z = 1.
(a) (4pts) Sketch the solid and its projection onto the xy-plane.
(b) (4pts) Suppose the density of the solid at the point
p P (x, y, z) is proportional to
the distance of P from the z-axis: δ(x, y, z) = C x2 + y 2 , where C is a constant.
Using symmetry, what are x̄ and ȳ?
(c) (8pts) Find M , the mass of the solid, using a triple integral in cylindrical coordinates.
(d) (4pts) Set up the triple integral for the first moment Mxy in cylindrical coordinates. Do not evaluate.
3. Consider the volume with outer surface ρ = 2 − cos φ and inner surface ρ = 1 (assume
0 ≤ θ ≤ 2π).
(a) (4pts) Sketch the cross section of the surface in the yz plane, shading appropriately.
(b) (10pts) Set up the triple integral representing volume using spherical coordinates.
(c) (4pts) Evaluate the integral.
4. Consider
F = hsin(yz) +
x
y
, xz cos(yz) +
, xy cos(yz) + ez i
2
2
1 + (xy)
1 + (xy)
(a) (9pts) Without finding the potential function, show that F is a conservative function.
(b) (6pts) Find a potential function f (x, y, z) corresponding to F.
π
(c) (2pts) Find the work done by F on a path going from
, 0, 1 to (1, 1, 0)
4
ZZ
5. Consider
x3 yexy dA where R is the region in the first quadrant bounded by the
R
curves y = 2/x, y = 4/x, y = 1/x3 , and y = 3/x3 . Let u = xy and v = x3 y be a change
of variables.
(a) (4pts) Verify that x = u−1/2 v 1/2 , y = u3/2 v −1/2 .
(b) (4pts) Write the equations corresponding to the boundary of R in terms of u, v.
(c) (4pts) Graph R in the xy plane and the region corresponding to R in the uv plane.
Please do not draw extremely tiny pictures. Define each vertex in the uv plane
using a coordinate point.
(d) (4pts) Compute the Jacobian J(u, v).
(e) (9pts) Set up, then evaluate, the integral in the new variables u and v.