MATH 317 Exam Review
1
Z
4
Z
4y−y 2
1. Reverse the order of integration in the integral
(y + 1) dx dy and evaluate.
−1
y−4
2. Rewrite the sum of iterated integrals as a single integral by reversing the order of
integration. Evaluate the integral.
Z 8 Z √y/3
Z 12 Z √y/3
y dx dy +
y dx dy.
√
0
0
Z
1
8
Z
1
3. Evaluate the integral
0
Z
1
Z
π
2
4. Evaluate
0
y−8
x2 sin(xy) dx dy.
y
ecos x dx dy.
sin−1 (y)
5. Evaluate the integral for f (x, y, z) = 2x−y +z over the region bounded by the cylinder
z = y 2 , the xy-plane, and the planes x = 0, x = 1, y = −2, y = 2.
6. Consider the iterated integral
Z
2
−2
Z
1
2
√
4−x2
Z
4−y 2
(x3 + y 3 ) dz dy dx.
x2 +3y 2
0
a) This integral is equal to a triple integral over a solid region W in R3 . Describe W.
b) Set up an equivalent iterated integral by integrating first with respect to x, then
with respect to z, then with respect to y.
7. Find the volume of the solid D bounded below by the paraboloid z = x2 + y 2 , and
above by the plane 2x + z = 3.
8. Find the volume bounded by the surface x2 + 3y 2 = z and the cylinder y 2 + z = 4.
9. Find the center of mass of the lamina of density δ(x, y) = x2 that occupies the region
R bounded by the parabola y = 2 − x2 and the line y = x.
Z √
Z Z √
1
1−x2
−1
√
− 1−x2
10. Evaluate the integral
coordinates.
2−x2 −y 2
z dz dy dx by converting to cylindrical
x2 +y 2
11. Find the volume of the region D bounded above by the paraboloid z = 4 − (x2 + y 2 ),
below by the plane z = 0, and laterally by the cylinder x2 + y 2 = 1.
12. Find the area of the region D in the xy-plane bounded by the line y = 14 x and y = 52 x,
and the hyperbolas xy = 1, and xy = 5 using the transformation u = xy , v = xy.
Z Z
2y−x
13. Evaluate
e y+2x dA, where R is the trapezoid with vertices (0, 2), (1, 0), (4, 0),
and (0, 8).
R
MATH 317 Exam Review
2
14. Use the change of variables x = ar cos θ, y = br sin θ to evaluate
2
Z Z
x
y2
exp − 2 − 2 dy dx
a
b
D
where D is the region bounded by the quarter ellipse
x2
a2
2
+ yb2 = 1 in the first quadrant.
15. Find divF and curlF for the vector field F = xyz i + x2 y 2 z 2 j + y 2 z 3 k.
Z
16. Evaluate (x2 + y 2 ) dx + 2xy dy for these choices of the curve C :
C
a) C is the quarter circle x2 + y 2 = 1 from (1, 0) to (0, 1).
b) C is the straight line y = 1 − x from (1, 0) to (0, 1).
I
17. Evaluate
(x2 y dx − xy dx) where C is the path that begins at (0, 0), goes to (1, 1)
C
along the parabola y = x2 , and then returns to (0, 0) along the line y = x.
18. Find the work done by the force field F = (x2 + y 2 ) i + (x + y) j as an object moves
counterclockwise along the circle x2 + y 2 = 1 from (1, 0) to (−1, 0) and back to (1, 0)
along the x-axis.
19. A 5, 000-lb satellite orbits the earth in a circular orbit 5, 000 mi from the center of the
earth. How much work is done as the satellite moves through one complete revolution?
20. Show that the vector field F = (ex sin y − y) i + (ex cos y − x − 2) j is conservative by
finding a scalar potential function for F.
R
21. Evaluate the line integral C F · dr where
F = [(2x − x2 y)e−xy + tan−1 y] i + [
y2
x
− x3 e−xy ] j
+1
for each of the following curves:
a) C1 : the ellipse 9x2 + 4y 2 = 36.
b) C2 : the curve with the parametric equations
x = t2 cos(πt), y = e−t sin(πt), 0 ≤ t ≤ 1.
I
y 2 dx + x2 dy where C is the path
22. Use Green’s theorem to calculate the line integral
C
formed by the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1) oriented counterclockwise.
I
23. Use Green’s theorem to evaluate the line integral
2y dx − x dy around the curve
C
C which starts at (2, 0), moves to (−2, 0) along the semicircle x2 + y 2 = 4, and then
returns to (2, 0) along the x-axis.
MATH 317 Exam Review
3
Z Z
(x2 + y 2 + z 2 ) dS where S is the portion of the plane z = x + 1 that lies
24. Evaluate
S
inside the cylinder x2 + y 2 = 1.
Z Z
25. Evaluate
F · dS where F = x i + 2y j − 3z k and S is the part of the plane
S
15x − 12y + 3z = 6 that lies above the unit square [0, 1] × [0, 1]. Assume that S is
oriented so that the normal vectors are directed outwards.
Z Z
26. Evaluate
F · dS where F = x2 i + z k and S is the parametric surface
S
x = sin u cosv, y = sin u sin v, z = cos u,
for 0 ≤ u ≤ π, 0 ≤ v ≤ 2π.
27. Find the area of the surface cut from the paraboloid z = 2x2 + 2y 2 by the planes z = 2
and z = 8.
28. Verify Stoke’s theorem implies Green’s theorem. Hint: In Stoke’s theorem take F =
P (x, y) i + Q(x, y) j + 0 k.
29. Let S be the surface defined by y = 10 − x2 − z 2 with y ≥ 1, oriented with rightwardpointing normal. Let
F = (2xyz + 5z) i + ex cos(yz) j + x2 y k.
Z Z
Determine
I
curlF · dS.
S
1
( y 2 dx + z dy + x dz) where C is the curve of intersection of the plane
C 2
x + z = 1, and the ellipsoid x2 + 2y 2 + z 2 = 1, oriented counterclockwise as viewed
from above.
I
31. Use Stoke’s theorem to evaluate
(z dx + x dy + y dz) where C is the triangle with
30. Evaluate
C
vertices (3, 0, 0), (0, 0, 2), and (0, 6, 0) traversed in this order.
Z Z
32. Use the divergence theorem to evaluate
F · n dS where F = xy i − z 2 k and S is
the surface of the upper five faces of the unit cube [0, 1] × [0, 1] × [0, 1].
Z Z
33. Use the divergence theorem to evaluate
krk(r · n) dS where r = x i + y j + z k
S
and S is the sphere x2 + y 2 + z 2 = a2 with constant a > 0.
MATH 317 Exam Review
4
Answers
1
625
12
2
896
15
3
1−sin 1
2
shadow in xy-plane
x2
4
176
15
6 a) Bottom surface z = x2 + 3y 2 , top surface z = 4 − y 2 ,
R 1 R 4−y2 R √z−3y2 3
2
√
(x + y 3 ) dx dz dy.
+ y ≤ 1, y ≥ 0, b) 0 3y2
2
4 e−1 5
−
(− 87 ,
2
√
7π 2
4
25 −6 26
7π
12
z−3y
1
7 8π 8 4π 9 (x, y) =
10
11
12 2 ln(10) 13 6(e2 − e− 2 ) 14 abπ
(1 − e−1 )
4
15 divF = yz + 2x2 yz + 3y z , curlF = (2yz − 2x2 y 2 z) i + xy j + (2xy 2 z 2 − xz) k
7
18 pi2 − 43 19 0 20 f (x, y) = ex sin y − xy − 2y 21 a) 0 b) 1
16 a) − 13 b) − 13 17 − 60
22 0 23 −6π 24
33 4πa4 .
− 20
)
49
2 2
4π
3
7π
2
3
3
27
3
(65 2 −17 2 )π
2π
28 29 45π 30 − π4 31 −18 32 − 12