MATH 234 EXAM 1 REVIEW PROBLEMS
Problem 1. Consider the integral
ˆ 0 ˆ √1+x
−1
ˆ
√
− 1+x
3
f (x, y) dydx +
0
ˆ
1−x
√
− 1+x
f (x, y) dydx.
(1) Sketch the region of integration.
(2) Rewrite the integral as one double integral with the order of integration reversed.
Problem 2. Consider the following iterated integral
ˆ 2ˆ 4
cos x2 dxdy.
I=
2y
0
Sketch the region on which this integral is defined, and evaluate the integral.
Problem 3. Let D be the region bounded by y = x, y = 4, x = 0. Set up the iterated integral for both
orders of integration:
¨
y 2 exy dA,
D
and evaluate using the easier order.
Problem 4. Compute
ˆ
ˆ
0
2y
4 − x2
−1/2
3/2
dxdy.
−1
Problem 5. Evaluate the following integral by converting to polar coordinates:
ˆ 1 ˆ √2−x2 p
x2 + y 2 dydx.
I=
x
0
Problem 6. Compute
ˆ
1
ˆ √1−y2
(1 − x) dxdy.
−1
0
Problem 7. Evaluate the following integral by converting to polar coordinates:
ˆ 3 ˆ √9−x2
sin x2 + y 2 dydx.
−3
0
Problem 8. Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides
of length a, where the density at any point is proportional to the square of the distance from the vertex
opposite the hypotenuse.
Problem 9. Find the volume of the region inside the sphere x2 + y 2 + z 2 = 25 and outside the cylinder
x2 + y 2 = 16.
Problem 10. A swimming pool is circular with a 40-ft diameter. The depth is constant along the east-west
lines, and increases linearly from 2 ft at the south end to 7 ft at the north end. Find the volume of the pool.
Problem 11. Express the following iterated integral
ˆ 1/2 ˆ √2x−x2 ˆ x+y+4
0
√
x2 + y 2 + z 2 dzdydx
3x
−x2 −y 2 −2
as an interated integral in cylindrical coordinates.
1
MATH 234 EXAM 1 REVIEW PROBLEMS
2
Problem 12. Consider the following triple integral:
ˆ 2 ˆ √2x−x2 ˆ √4−x2 −y2
√
0
−
0
dzdydx.
4−x2 −y 2
(1) Describe the solid region of integration by giving the equations of the surfaces that bound the solid.
(2) Convert the triple integral into an iterated integral in cylindrical coordinates, but do not evaluate.
˝
Problem 13. Express the triple integral D xdV as an iterated integral in spherical coordinates, where D
is the solid region bounded above by the sphere x2 + y 2 + z 2 = 2z and below by the plane z = 1. (Do not
evaluate).
Problem 14. Suppose T is a transformation of the uv plane to the xy plane and we know that in the limit
∆u, ∆v → 0, both the rectangle with lower left corner (0, −2) width ∆u and height ∆v, and the rectangle
with lower left corner (5, 2) width ∆u and height ∆v map to patches of eight times their original areas.
(1) Which two of the following transformations
could be T ?
(a) x = u/3, y = v 12 + v 2
(b) x = u2 /3, y = v (2v − 1)
(c) x = v (v + 2), y = u/3
(d) x = 4v 3 , y = −u/6
(2) Could you differentiate between the two transformations from (1) if you knew how T changed the
area of the rectangle with lower left corner (u0 , v0 ), where v0 is equal to neither −2 or 2? Why/Why
not?
Problem 15. Compute the integral
¨
y 2 dA,
D
where D is the region in the first quadrant bounded by the curves xy = 1, y = x, y = 4x, and xy = 2.
Problem 16. Compute the integral
¨
x + 2y
dA,
3/2
(3x − 2y)
where D is the region bounded by the curves x + 2y = 0, x + 2y = 2, 3x − 2y = 4, and 3x − 2y = 1.
D
Problem 17. Sketch the region of integration and evaluate the double integral
ˆ 4 ˆ y/2+1
(2x − y) sin (2x − y) dxdy.
0
y/2
Problem 18. Use a change of variables to express the following integral as an integral over the unit sphere:
˚ p
2
y2
z2
− x
2 + b 2 + c2
a
x2 + y 2 e
dV,
E
where
E=
(Do not evaluate).
x2
y2
z2
(x, y, z) : 2 + 2 + 2 ≤ 1 .
a
b
c