Name:
Practice Midterm
1. Compute the following double integral by switching the order of integration
ˆ 1ˆ 1
x3
p
dydx.
x4 + y 2
0
x2
(Hint: sketch the domain of integration first.)
2. Compute the integral
˚
x dV,
E
where E is the tetrahedron with vertices (0, 0, 0), (2, 0, 0), (0, 1, 0), (0, 0, 1).
3. Compute the integral
ˆ
2
ˆ √4−y2
−2
e−(x
2 +y 2 )
dxdy,
0
by switching p
to polar coordinates. Correction! The upper bound on the x-coordinate was
it should be 4 − y 2 .
4. Evaluate the line integral
p
1 − y2,
ˆ
F~ · d~r,
C
where F~ (x, y, z) = hy 3 , 3xy 2 + ez , yez i, and the curve C is parametrized by
~r(t) = het , t5 , ln(1 + ln(t10 − t5 + 1))i,
with 0 6 t 6 1. (Hint: there is an easy way and a hard way to do this problem. Don’t do it the
hard way!)
5. Calculate the work done by the vector field F~ (x, y) = hxy, 3y 2 i on a particle moving along the
path ~r(t) = h11t4 , t3 i from t = 0 to t = 1.
˚
6. Evaluate
xyz dV,
E
where E is the region in the first octant with x2 + y 2 + z 2 6 4 (recall that in the first octant we
have x > 0, y > 0, z > 0).
7. Evaluate
ffi
y 4 dx + 2xy 3 dy,
C
where C is the ellipse x2 + 2y 2 = 2. (Hint: use Green’s theorem.)
1