Math 6A Practice Problems II
Written by Victoria Kala
vtk[email protected]
SH 6432u Office Hours: R 12:30 − 1:30pm
Last updated 5/13/2016
1. Evaluate the line integral
R
C
xyzds where C is the curve parametrized by x = 2 sin t, y =
t, z = −2 cos t, 0 ≤ t ≤ π.
2. Evaluate the line integral
R
C
F ·dr where C is given by the vector function r(t) = ti + sin tj +
cos tk, 0 ≤ t ≤ π and F = zi + yj − xk.
3. Determine whether or not F is a conservative vector field. If it is, find a function f such that
F = ∇f .
(a) F(x, y) = e
x
cos yi + e
x
sin yj
(b) F(x, y) = (ln y + 2xy
3
)i +
3x
2
y
2
+
x
y
j
4. Let F(x, y, z) = y
2
cos zi + 2xy cos zj − xy
2
sin zk, C be the curve parametrized by r(t) =
t
2
i + sin tj + tk, 0 ≤ t ≤ π.
(a) Show that F is conservative. (Hint: use curl)
(b) Find a function f such that F = ∇f .
(c) Use (b) to calculate
R
C
F · dr along the given curve C.
5. (a) Estimate the volume of the solid that lies below the surface z = x + 2y
2
and above the
rectangle R = [0, 2] ×[0, 4]. Use a Riemann sum with m = n = 2 and choose the sample
points to be the lower right corners.
(b) Use the midpoint rule to estimate the volume in (a).
(c) Calculate the exact volumes of the solid.
6. Evaluate the following integrals:
(a)
Z
4
1
Z
2
1
x
y
+
y
x
dydx
(b)
Z
1
0
Z
1
0
√
s + t dsdt
7. Evaluate
RR
D
(x + y) dA where D is bounded by y =
√
x, y = x
2
.
8. Evaluate the integral by reversing the order of integration:
Z
1
0
Z
π/2
arcsin y
cos x
p
1 + cos
2
x dxdy.
1
9. Evaluate
RR
R
(x + y) dA where R is the region that lies to the left of the y-axis between the
circles x
2
+ y
2
= 1, x
2
+ y
2
= 4.
10. Verify Green’s Theorem for
R
C
x
4
dx + xydy where C is the triangular curve consisting of line
segments from (0, 0) to (1, 0), (1, 0), to (0, 1), and (0, 1) to (0, 0) traversed in that order.
11. Evaluate
R
C
y
2
dx + 3xydy, where C is the boundary of the semiannular region D in the upper
half plane between the circles x
2
+ y
2
= 1 and x
2
+ y
2
= 4. (You may assume that C is
positively oriented.)
12. Use a triple integral to find the volume of the solid bounded by the cylinder y = x
2
and the
planes z = 0, z = 4, and y = 9.
13. Evaluate
RRR
E
xydV where E is bounded by the parabolic cylinders y = x
2
and x = y
2
, and
the planes z = 0 and z = x + y.
14. Evaluate
RRR
E
(x
3
+ xy
2
)dV where E is the solid in the first octant that lies beneath the
paraboloid z = 1 − x
2
− y
2
.
15. Evaluate
RRR
E
e
z
dV where E is enclosed by the paraboloid z = 1 + x
2
+ y
2
, the cylinder
x
2
+ y
2
= 5, and the xy-plane.
16. Evaluate
RRR
E
xyzdV where E lies between the spheres ρ = 2 and ρ = 4 and the cone φ =
π
3
.
2