Math 233.
Practice Questions
1. Consider the sum of double integrals as follows:
Z 10 Z
Z 3Z 6
dy dx +
−4
2−x
3
Chapter 14
6
dy dx
x−4
(a) This sum represents the area of a region; sketch the region.
(b) Determine the area of the region.
(c) Rewrite the integral in the order dx dy (it should be only one double integral, not a sum).
(d) Evaluate the integral you found in (c) to verify it is equal to the area of the region.
2. A planar region is bounded by the line y = x − 2 and the parabola y = x2 − 5x − 18. Set
up but do not evaluate a double integral to represent the area of the region.
3. Find a function f (x, y) such that
∂f
= 5e5x cos(6y) − 8x
∂x
Z
5
Z
x4
4. Evaluate the iterated integral
0
0
and
∂f
= 6y − 6e5x sin(6y),
∂y
1 x9
ye dy dx
8
5. Consider the region in the first quadrant bounded by the graphs of
x = y3
and
4y = x
Express the area of the region as iterated integrals in both orders.
6. Find the volume of the solid region that lies below the graph z = 16 − y 2 and above the
triangular region R in the xy-plane bounded by the y-axis and the lines y = x and y = 4.
7. Sketch the region of integration for
Z 2 ln 3 Z
3
ex/2
0
4
dy dx
ln y
and then evaluate the integral, switching the order of integration, as necessary.
Z
4
Z
1
8. Evaluate the double integral
0
y/4
2
ex dx dy by switching the order of integration.
9. Evaluate the following integral by converting it to polar coordinates
Z 0Z 0
p
3
x2 + y 2 dy dx
√
− 36−x2
−6
p
10. Find the volume of the solid enclosed by z = 2 − x2 − y 2 and z = 1 x2 + y 2 .
11. Find the volume of the solid region bounded above by the graph
p
z = 49 − x2 − y 2
√
and below by the region R in the xy-plane given by 0 ≤ y ≤ 9 − x2 , −3 ≤ x ≤ 3.
12. Evaluate the following integral by converting it to polar coordinates
Z Z √
8−y 2
2
6(x2 + y 2 )1/2 dx dy
0
y
13. Set-up, but do not evaluate, the integrals to find M , Mx and My for the planar lamina
bounded by
y = 3 − x2
and
y = −3x − 1
whose density is per unit area given by the density function δ(x, y) = 3y 2 . Make sure to clearly
identify which integral is which.
14. A planar lamina in the xy-plane is given by
√
and
0 ≤ y ≤ 25 − x2
p
and its density is given by ρ(x, y) = 36 − x2 − y 2 .
−5≤x≤5
(a) Use a double integral to find the mass of the lamina.
(b) Set-up the integrals needed to find Mx and My for the lamina.
Z
1
Z
2
15. Interpret
−1
yex
4 +y 4
dx dy as a moment Mx . Describe the region for which it is a
−2
moment, and the density function. What do you expect Mx to be?
16. Given that f (x, y) is a continuously differentiable function, write the formula for dS, and
the integral formula for the surface area of z = f (x, y) over a region R in the xy-plane.
17. Find the surface area of the portion of the graph
z = 4x + 2y + 6
that lies above the triangle with vertices (0, 0), (6, 0), (6, 2) in the xy-plane.
18. A cylindrical drill of radius 10 units is used to bore a hole through the center of a solid
ball of radius 23 units.
(a) Use a double integral in polar coordinates to find the volume of the annular shaped solid
that remains.
(b) Use a double integral to find the surface area of the outer surface of the object.
19.
Find the surface area of the portion of the graph z = 36 + x2 − y 2 that lies above the
region R = {(x, y) : 4 ≤ x2 + y 2 ≤ 25}.
20. Use a double integral to find the surface area of the portion of the cone z = 3
that lies between the planes z = 3 and z = 12.
p
x2 + y 2
21. Assume the solid bounded by z = 0 and z = 64 − x2 − y 2 has constant density δ = k.
Find the centroid (x̄, ȳ, z̄) of the solid.
Hint. Note that x̄ = ȳ = 0 by symmetry, so it remains to find z̄ using the formula
Z Z Z
Z Z Z
Mxy
z̄ =
where Mxy =
zδ(x, y, z)dV, M =
δ(x, y, z)dV
M
Q
Q
you may wish to convert your integrals to polar coordinates once they have been reduced to
double integrals.
22. Use a triple integral to find the volume of the solid bounded by
x = 49 − y 2 , z = 0, z = 8x
23. Set-up triple integrals in rectangular coordinates that represent the volume of the solid
enclosed by the planes x = 0, y = 0, z = 0 and z = 16 − 4x − 4y using: (a) the order dz dy dx;
and (b) the order dy dx dz. Do not evaluate the integrals.
24. Assume the solid bounded by z = 0 and z = 100−x2 −y 2 has constant density δ(x, y, z) = k.
Use cylindrical coordinates to find the centroid (x̄, ȳ, z̄) of the solid.
Hint. Note that x̄ = ȳ = 0 by symmetry, so it remains to find z̄ using the formula
Z Z Z
Z Z Z
Mxy
where Mxy =
zδ(x, y, z)dV, M =
δ(x, y, z)dV
z̄ =
M
Q
Q
Z
2π
Z
π
Z
25. (a) The triple integral in spherical coordinates
0
volume of a solid. Describe the solid.
0
6
ρ2 sin φ dρ dφ dθ represents the
3
(b) Set-up but do not evaluate an integral in cylindrical coordinates to find the volume of a
right circular cylinder of height 2 units and radius 4 units.
26. Convert the integral
Z
3
−3
√
9−x2
Z
Z √9−x2 −y2
0
x2
0
5x
dz dy dx
+ + z2 + 3
y2
to (a) cylindrical coordinates and (b) spherical coordinates. Do not evaluate the integral.
27. Convert the integral
Z
9
−9
Z
√
81−x2
√
− 81−x2
√
Z
9+
81−x2 −y 2
x dz dy dx
9
to both cylindrical and spherical coordinates, and evaluate the integral.
p
28. Find the volume of the solid that lies above the cone z = 2 x2 + y 2 and between the
spheres x2 + y 2 + z 2 = 42 and x2 + y 2 + z 2 = 62
29. Let R be triangular region with vertices (0, 0), (−3, 1) and (1, 1). Use the change of variable
u = x + 3y and v = x − y and find the image S in the uv-plane under this transformation.
(a) Find the Jacobian
∂(x, y)
.
∂(u, v)
(b) Sketch R in the xy-plane and S in the uv-plane.
30. Consider the double integral
Z Z
R
14
p
dA
(2x − y)(5x + y)
where R is the region described by
4 ≤ 2x − y ≤ 16
and
1 ≤ 5x + y ≤ 4
Use the change of variable u = 2x − y and v = 5x + y to do the following:
(a) Find x and y in terms of u and v.
(b) Find the Jacobian
∂(x, y)
.
∂(u, v)
(c) Describe the region S in the uv-plane under this change of variable.
(d) Use the change of variable theorem to evaluate the given double integral.
3
1
31. Let R be the region bounded by y = , y = , x = 1 and x = 5. Use a change of variable
x
x
to evaluate the integral
Z Z
15x4 y 4
dA
1 + x5 y 5
R
To aid your solution, sketch the region R in the xy-plane, and the region S in the uv-plane.
32. Use the change of variable u = x + y and v = x − y to evaluate the double integral
Z Z
(x + y)ex−y dA
R
where R is the tilted rectangle with vertices (3, 1), (5, 3), (6, 2) and (4, 0).
Note: the region R is bounded by the lines x − y = 2, x − y = 4, x + y = 4 and x + y = 8