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(9/7/08)
Math 20C. Lecture Examples.
Section 15.3. Triple integrals†
Example 1 What is the geometric significance of
V
1 dx dy dz?
Answer:
V
1 dx dy dz = [Volume of V ]
Example 2 Express
V
4 xyz dx dy dz as an iterated integral of the form,
R
z=h(x,y)
z=g(x,y)
f (x, y, z) dz
dx dy
where V is the box defined by 0 ≤ x ≤ 3 , 0 ≤ y ≤ 2 , 0 ≤ z ≤ 1.
Answer:
V
4 xyz dx dy dz =
R
∫ z=
z=
4 xyz dz dy dx with R: 0 ≤ x ≤ 3 , 0 ≤ y ≤ 2 in Figure A2.
3 x
y
2
R
Figure A
Example 3 Evaluate the integral
V
4 xyz dx dy dz =
R
∫ z=
z=
4 xyz dz dy dx from
Example 2, where R is the rectangle in Figure A2.
Answer: Figure A3 •
V
4 xyz dx dy dz = 18
3 x
y
2
x
y = 2
R
Figure A
†Lecture notes to accompany Section 15.3 of Calculus, Early Transcendentals by Rogawski.
Math 20C. Lecture Examples. (9/7/08) Section 15.3, p. 2
Example 4 The solid V in xyz-space with distances measured in meters is bounded by z = 0, z = y, y = x^2 , and y = 1. Its density at (x, y, z) is ρ(x, y, z) = 8yz kilograms per cubic meter. (a) Express the mass of V as an iterated integral. (b) Evaluate the integral.
Answer: (a) Figure A4 • [Mass] =
V
ρ(x, y, z) dx dy dz =
R
∫ z=y
z=
8 yz dx dy dz
=
∫ x=
z=− 1
∫ y=
y=x^2
∫ z=y
z=
8 yz dz dy dx (b) [Total charge] = 169 kilograms
− 1 1 x
y
x
y = 1
R (^) y = x^2
Figure A
Example 5 What is the average value of g(x, y, z) = xey^ sin z on the cube V : 0 ≤ x ≤ 2 , 0 ≤ y ≤ 2 , 0 ≤ z ≤ 2? Answer: [Average value] = 14 (e^2 − 1)[1 − cos(2)]
