(^700) Iterated and multiple integrals
and the parallel axis theorem (Theorem 13.48) hold for solidsas well as
for lamina.
Problems 13.69
1 Set up and evaluate a threefold iterated integral for the volume Y of the
solid tetrahedron bounded by the coordinate planes and the plane having the
equation
a++b c-1,
it being assumed that a, b, c are positive numbers. As.:
Y - fax fbcl-?a/ f c1
z &
° b)dz = gabc.
(^2) Supposing that f is continuous over the tetrahedron T of the preceding
problem, set up a threefold iterated integral equal to the triple integral
fTf
(x,y,z) dx dy dz,
where, as is often done when rectangular coordinates are involved, dx dy dz is
written instead of dT. 11ns.: Same as answer to preceding problem except that
the integrand is f(x,y,z) instead of 1.
3 Set up a threefold iterated integral for the volume V of the solid in Ez
.bounded by the parabolic cylinders having the equations y = x2 and x = y2
and the planes having the equations z = 0 and x + y + z = 2. .dns.:
Y
=foI dx JO2-z-y A.
4 A homogeneous cube has density S and has edges of length a. Find its
moment of inertia about an edge. 41ns.: %8a5 or *Ma2, where M is the mass
of the cube.
5 A long solid circular cylinder S of radius b has its axis on the y axis ofan
x, y, z coordinate system. A circular hole having radius a and having its axis
on the z axis is drilled. Supposing that 0 < a 5 b, set up an integral for the
volume Y of the part of S that is drilled away. 4ns.: Because of symmetry,
V is 8 times the volume of the part in the first octant and
Y=8 (dx dy
o
'dz.
JO JO
(^6) Let q be a nonnegative constant and let S be a spherical ball of radius R
whose density is proportional to the qth power of the distance from thecenter.
Taking the origin at the center of the ball, setup a triple integral for the polar
moment of inertia of the ball about the z axis. Simplify matters by using the
fact that the total moment is 8 times the moment of thepart of the ball in the
first octant. Ans.:
t R R'-ii R'-z=-y=
lu
(lu)
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