James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Cut here and keep for reference
Chapter 15
(d) What is a type 1 solid region? How do you evaluate
yyy E
f sx, y, zd dV if E is such a region?
A region E is of type 1 if it lies between the graphs of two
continuous functions of x and y, that is,
E − 5sx, y, zd | sx, yd [ D, u1sx, yd < z < u2sx, yd6
where D is the projection of E onto the xy-plane. Then
y y f sx, y, zd dV − y Fy
E
D
u2sx, yd
u1sx, yd
f sx, y, zd dzG dA
(e) What is a type 2 solid region? How do you evaluate
yyy E
f sx, y, zd dV if E is such a region?
A type 2 region is of the form
Concept Check Answers (continued)
E − 5sx, y, zd | sy, zd [ D, u1sy, zd < x < u2sy, zd6
where D is the projection of E onto the yz-plane. Then
8. Suppose a solid object occupies the region E and has density
function sx, y, zd. Write expressions for each of the
following.
(a) The mass:
m − y y sx, y, zd dV
E
(b) The moments about the coordinate planes:
M yz − y y x sx, y, zd dV
E
M xz − y y y sx, y, zd dV
E
M x y − y y z sx, y, zd dV
E
(c) The coordinates of the center of mass:
y y f sx, y, zd dV − y Fy
E
D
u2sy, zd
u1sy, zd
f sx, y, zd dxG dA
(f) What is a type 3 solid region? How do you evaluate
yyy E
f sx, y, zd dV if E is such a region?
A type 3 region is of the form
E − 5sx, y, zd | sx, zd [ D, u1sx, zd < y < u2sx, zd6
where D is the projection of E onto the xz-plane. Then
y y f sx, y, zd dV − y Fy
E
D
u2sx, zd
u1sx, zd
f sx, y, zd dyG dA
sx, y, zd, where x − Myz
m , y − Mxz
m , z − Mxy
m
(d) The moments of inertia about the axes:
I x − y y sy 2 1 z 2 d sx, y, zd dV
E
I y − y y sx 2 1 z 2 d sx, y, zd dV
E
I z − y ysx 2 1 y 2 d sx, y, zd dV
E
(continued)
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