CalculusVolume3-OP_mktoy8b-115

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212. The solid E bounded by y = x, x = 4, y = 0,
and z = 1 is given in the following figure. Evaluate the
integral ∭
E
xyz dV by integrating first with respect to x,
then y, and then z.
213. [T] The volume of a solid E is given by the integral
∫
−2
0
∫
x
0
∫
0
x2 + y2
dz dy dx. Use a computer algebra system
(CAS) to graph E and find its volume. Round your answer
to two decimal places.
214. [T] The volume of a solid E is given by the integral
∫
−1
0
∫
−x2
0
∫
0
1 + x2 + y2
dz dy dx. Use a CAS to graph E and
find its volume V . Round your answer to two decimal
places.
In the following exercises, use two circular permutations
of the variables x, y, and z to write new integrals whose
values equal the value of the original integral. A circular
permutation of x, y, and z is the arrangement of the
numbers in one of the following orders:
y, z, and x or z, x, and y.
215. ∫
0
1
∫
1
3
∫
2
4
⎛
⎝x2 z2 + 1⎞
⎠dx dy dz
216. ∫
1
3
∫
0
1
∫
0
−x + 1
⎛
⎝2x + 5y + 7z⎞
⎠dy dx dz
217. ∫
0
1
∫
−y
y
∫
0
1 − x4 − y4
ln x dz dx dy
218. ∫
−1
1
∫
0
1
∫
−y6
y
(x + yz)dx dy dz
219. Set up the integral that gives the volume of the solid
E bounded by y2 = x2 + z2 and y = a2, where a > 0.
220. Set up the integral that gives the volume of the solid
E bounded by x = y2 + z2 and x = a2, where a > 0.
221. Find the average value of the function
f (x, y, z) = x + y + z over the parallelepiped determined
by x = 0, x = 1, y = 0, y = 3, z = 0, and z = 5.
222. Find the average value of the function
f (x, y, z) = xyz over the solid
E = [0, 1] × [0, 1] × [0, 1] situated in the first octant.
223. Find the volume of the solid E that lies under the
plane x + y + z = 9 and whose projection onto the xy
-plane is bounded by x = y − 1, x = 0, and x + y = 7.
224. Find the volume of the solid E that lies under the
plane 2x + y + z = 8 and whose projection onto the xy
-plane is bounded by x = sin−1 y, y = 0, and x = π
2.
225. Consider the pyramid with the base in the xy -plane
of [−2, 2] × [−2, 2] and the vertex at the point (0, 0, 8).
a. Show that the equations of the planes of the lateral
faces of the pyramid are 4y + z = 8,
4y − z = −8, 4x + z = 8, and −4x + z = 8.
b. Find the volume of the pyramid.
226. Consider the pyramid with the base in the xy -plane
of [−3, 3] × [−3, 3] and the vertex at the point (0, 0, 9).
a. Show that the equations of the planes of the side
faces of the pyramid are 3y + z = 9,
3y + z = 9, y = 0 and x = 0.
b. Find the volume of the pyramid.
Chapter 5 | Multiple Integration 563
227. The solid E bounded by the sphere of equation
x2 + y2 + z2 = r2 with r > 0 and located in the first
octant is represented in the following figure.
a. Write the triple integral that gives the volume of
E by integrating first with respect to z, then with
y, and then with x.
b. Rewrite the integral in part a. as an equivalent
integral in five other orders.
228. The solid E bounded by the equation
9x2 + 4y2 + z2 = 1 and located in the first octant is
represented in the following figure.
a. Write the triple integral that gives the volume of
E by integrating first with respect to z, then with
y, and then with x.
b. Rewrite the integral in part a. as an equivalent
integral in five other orders.
229. Find the volume of the prism with vertices
(0, 0, 0), (2, 0, 0), (2, 3, 0),
(0, 3, 0), (0, 0, 1), and (2, 0, 1).
230. Find the volume of the prism with vertices
(0, 0, 0), (4, 0, 0), (4, 6, 0),
(0, 6, 0), (0, 0, 1), and (4, 0, 1).
231. The solid E bounded by z = 10 − 2x − y and
situated in the first octant is given in the following figure.
Find the volume of the solid.
232. The solid E bounded by z = 1 − x2 and situated
in the first octant is given in the following figure. Find the
volume of the solid.
564 Chapter 5 | Multiple Integration
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233. The midpoint rule for the triple integral
∭
B
f (x, y, z)dV over the rectangular solid box B is a
generalization of the midpoint rule for double integrals.
The region B is divided into subboxes of equal sizes and
the integral is approximated by the triple Riemann sum
∑
i = 1
l
∑
j = 1
m
∑
k = 1
n
f ⎛
⎝ xi
– , y j
– , zk
– ⎞
⎠ΔV , where ⎛
⎝ xi
– , y j
– , zk
– ⎞
⎠ is
the center of the box Bi jk and ΔV is the volume of
each subbox. Apply the midpoint rule to approximate
∭
B
x2 dV over the solid
B = ⎧
⎩
⎨(x, y, z)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1⎫
⎭
⎬ by using
a partition of eight cubes of equal size. Round your answer
to three decimal places.
234. [T]
a. Apply the midpoint rule to approximate
∭
B
e−x2
dV over the solid
B = ⎧
⎩
⎨(x, y, z)|0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1⎫
⎭
⎬
by using a partition of eight cubes of equal size.
Round your answer to three decimal places.
b. Use a CAS to improve the above integral
approximation in the case of a partition of n3
cubes of equal size, where n = 3, 4,…, 10.
235. Suppose that the temperature in degrees Celsius at
a point (x, y, z) of a solid E bounded by the coordinate
planes and x + y + z = 5 is T(x, y, z) = xz + 5z + 10.
Find the average temperature over the solid.
236. Suppose that the temperature in degrees Fahrenheit
at a point (x, y, z) of a solid E bounded by the coordinate
planes and x + y + z = 5 is T(x, y, z) = x + y + xy.
Find the average temperature over the solid.
237. Show that the volume of a right square pyramid of
height h and side length a is v = ha2
3 by using triple
integrals.
238. Show that the volume of a regular right hexagonal
prism of edge length a is 3a3 3
2 by using triple
integrals.
239. Show that the volume of a regular right hexagonal
pyramid of edge length a is a3 3
2 by using triple
integrals.
240. If the charge density at an arbitrary point (x, y, z)
of a solid E is given by the function ρ(x, y, z), then the
total charge inside the solid is defined as the triple integral
∭
E
ρ(x, y, z)dV . Assume that the charge density of the
solid E enclosed by the paraboloids x = 5 − y2 − z2 and
x = y2 + z2 − 5 is equal to the distance from an arbitrary
point of E to the origin. Set up the integral that gives the
total charge inside the solid E.
Chapter 5 | Multiple Integration 565
5.5 | Triple Integrals in Cylindrical and Spherical
Coordinates
Learning Objectives
5.5.1 Evaluate a triple integral by changing to cylindrical coordinates.
5.5.2 Evaluate a triple integral by changing to spherical coordinates.
Earlier in this chapter we showed how to convert a double integral in rectangular coordinates into a double integral in polar
coordinates in order to deal more conveniently with problems involving circular symmetry. A similar situation occurs with
triple integrals, but here we need to distinguish between cylindrical symmetry and spherical symmetry. In this section we
convert triple integrals in rectangular coordinates into a triple integral in either cylindrical or spherical coordinates.
Also recall the chapter opener, which showed the opera house l’Hemisphèric in Valencia, Spain. It has four sections with
one of the sections being a theater in a five-story-high sphere (ball) under an oval roof as long as a football field. Inside is
an IMAX screen that changes the sphere into a planetarium with a sky full of 9000 twinkling stars. Using triple integrals
in spherical coordinates, we can find the volumes of different geometric shapes like these.
Review of Cylindrical Coordinates
As we have seen earlier, in two-dimensional space ℝ2, a point with rectangular coordinates (x, y) can be identified
with (r, θ) in polar coordinates and vice versa, where x = r cos θ, y = r sin θ, r2 = x2 + y2 and tan θ = ⎛
⎝
y
x
⎞
⎠ are the
relationships between the variables.
In three-dimensional space ℝ3, a point with rectangular coordinates (x, y, z) can be identified with cylindrical
coordinates (r, θ, z) and vice versa. We can use these same conversion relationships, adding z as the vertical distance to
the point from the xy -plane as shown in thefollowing figure.
Figure 5.50 Cylindrical coordinates are similar to polar
coordinates with a vertical z coordinate added.
To convert from rectangular to cylindrical coordinates, we use the conversion x = r cos θ and y = r sin θ. To convert
from cylindrical to rectangular coordinates, we use r2 = x2 + y2 and θ = tan−1 ⎛
⎝
y
x
⎞
⎠. The z -coordinate remains the same
in both cases.
In the two-dimensional plane with a rectangular coordinate system, when we say x = k (constant) we mean an unbounded
vertical line parallel to the y -axis and when y = l (constant) we mean an unbounded horizontal line parallel to the x -axis.
566 Chapter 5 | Multiple Integration
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With the polar coordinate system, when we say r = c (constant), we mean a circle of radius c units and when θ = α
(constant) we mean an infinite ray making an angle α with the positive x -axis.
Similarly, in three-dimensional space with rectangular coordinates (x, y, z), the equations x = k, y = l, and z = m,
where k, l, and m are constants, represent unbounded planes parallel to the yz -plane, xz -plane and xy -plane,
respectively. With cylindrical coordinates (r, θ, z), by r = c, θ = α, and z = m, where c, α, and m are constants,
we mean an unbounded vertical cylinder with the z -axis as its radial axis; a plane making a constant angle α with
the xy -plane; and an unbounded horizontal plane parallel to the xy -plane, respectively. This means that the circular
cylinder x2 + y2 = c2 in rectangular coordinates can be represented simply as r = c in cylindrical coordinates. (Refer to
Cylindrical and Spherical Coordinates for more review.)
Integration in Cylindrical Coordinates
Triple integrals can often be more readily evaluated by using cylindrical coordinates instead of rectangular coordinates.
Some common equations of surfaces in rectangular coordinates along with corresponding equations in cylindrical
coordinates are listed in Table 5.1. These equations will become handy as we proceed with solving problems using triple
integrals.
Circular cylinder Circular cone Sphere Paraboloid
Rectangular x2 + y2 = c2 z2 = c2 ⎛
⎝x2 + y2⎞
⎠ x2 + y2 + z2 = c2 z = c⎛
⎝x2 + y2⎞
⎠
Cylindrical r = c z = cr r2 + z2 = c2 z = cr2
Table 5.1 Equations of Some Common Shapes
As before, we start with the simplest bounded region B in ℝ3, to describe in cylindrical coordinates, in the form
of a cylindrical box, B = ⎧
⎩
⎨(r, θ, z)|a ≤ r ≤ b, α ≤ θ ≤ β, c ≤ z ≤ d⎫
⎭
⎬ (Figure 5.51). Suppose we divide each interval
into l, m and n subdivisions such that Δr = b − a
l , Δθ = β − α
m , and Δz = d − c
n . Then we can state the following
definition for a triple integral in cylindrical coordinates.
Figure 5.51 A cylindrical box B described by cylindrical
coordinates.
Chapter 5 | Multiple Integration 567
	Chapter 5. Multiple Integration
	5.5. Triple Integrals in Cylindrical and Spherical Coordinates*