Calculus 2nd Edition Rogawski

942 C H A P T E R 16 MULTIPLE INTEGRATION
∫∫
24. Evaluate xdA, where D is the shaded domain in Figure 2.
D
2
y
r = 2(1 + cos θ)
D
∫∫
33. Use polar coordinates to calculate
D
√
x 2 + y 2 dA, where D is
the region in the first quadrant bounded by the spiral r = θ, the circle
r = 1, and the x-axis.
∫∫
34. Calculate sin(x 2 + y 2 )dA, where
D
{ π
D =
2 ≤ x2 + y 2 ≤ π}
2 4
FIGURE 2
x
35. Express in cylindrical coordinates and evaluate:
∫ 1 ∫ √ 1−x 2 ∫ √ x 2 +y 2
zdzdydx
0 0 0
25. Find the volume of the region between the graph of the function
f (x, y) = 1 − (x 2 + y 2 ) and the xy-plane.
∫ 3 ∫ 4 ∫ 4
26. Evaluate (x 3 + y 2 + z) dx dy dz.
0 1 2
∫∫∫
27. Calculate (xy + z) dV , where
B
B = { 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, 1 ≤ z ≤ 3 }
as an iterated integral in two different ways.
∫∫∫
28. Calculate xyz dV , where
W
W = { 0 ≤ x ≤ 1, x≤ y ≤ 1, x≤ z ≤ x + y }
∫ 1 ∫ √ 1−x 2
29. Evaluate I =
−1 0
∫ 1
(x + y + z) dz dy dx.
0
30. Describe a region whose volume is equal to:
∫ 2π ∫ π/2 ∫ 9
(a)
ρ 2 sin φ dρ dφ dθ
0 0 4
∫ 1 ∫ π/4 ∫ 2
(b)
rdrdθ dz
−2 π/3 0
∫ 2π ∫ 3 ∫ 0
(c)
√ rdzdrdθ
0 0 − 9−r 2
31. Find the volume of the solid contained in the cylinder x 2 + y 2 = 1
below the curve z = (x + y) 2 and above the curve z = −(x − y) 2 .
∫∫
32. Use polar coordinates to evaluate xdA, where D is the shaded
D
region between the two circles of radius 1 in Figure 3.
1
y
36. Use spherical coordinates to calculate the triple integral of
f (x, y, z) = x 2 + y 2 + z 2 over the region
1 ≤ x 2 + y 2 + z 2 ≤ 4
37. Convert to spherical coordinates and evaluate:
∫ 2 ∫ √ 4−x 2 ∫ √ 4−x 2 −y 2
√ e −(x2 +y 2 +z 2 ) 3/2 dzdy dx
−2 − 4−x 2 0
38. Find the average value of f (x, y, z) = xy 2 z 3 on the box [0, 1] ×
[0, 2] × [0, 3].
39. Let W be the ball of radius R in R 3 centered at the origin, and let
P = (0, 0,R)be the North Pole. Let d P (x, y, z) be the distance from
P to (x, y, z). Show that the average value of d P over the sphere W is
equal to d = 6R/5. Hint: Show that
d = 1 ∫ 2π ∫ R ∫ π √
ρ 2 sin φ R
4
2 + ρ 2 − 2ρR cos φ dφ dρ dθ
3 πR3 θ=0 ρ=0 φ=0
and evaluate.
40. Express the average value of f (x, y) = e xy over the ellipse
x2
2 + y2 = 1 as an iterated integral, and evaluate numerically
using a computer algebra system.
41. Use cylindrical coordinates to find the mass of the solid bounded
by z = 8 − x 2 − y 2 and z = x 2 + y 2 , assuming a mass density of
f (x, y, z) = (x 2 + y 2 ) 1/2 .
42. Let W be the portion of the half-cylinder x 2 + y 2 ≤ 4,x ≥ 0 such
that 0 ≤ z ≤ 3y. Use cylindrical coordinates to compute the mass of
W if the mass density is ρ(x, y, z) = z 2 .
43. Use cylindrical coordinates to find the mass of a cylinder of radius
4 and height 10 if the mass density at a point is equal to the square of
the distance from the cylinder’s central axis.
1
x
44. Find the centroid of the region W bounded, in spherical coordinates,
by φ = φ 0 and the sphere ρ = R.
FIGURE 3
45. Find the centroid of the solid bounded by the xy-plane, the cylinder
x 2 + y 2 = R 2 , and the plane x/R + z/H = 1.