13.7 Triple integrals; cylindrical coordinates 705(^4) Assuming that the solid cone shown in the upper part of Figure 13.791
has density (mass per unit volume) 5(p,q5,z) at the point having cylindrical coordi-
nates (p,cb,z), set up an iterated integral for the moment of inertia of the cone
Figure 13.791
about the x axis. Then calculate the required moment for the case in which the
density is proportional to the qth power of the distance from the axis of the cone,
that is, S(p,¢,z) = kpa. dns.: The integral is the same as that in (13.77) except
that the lower limit of integration with respect to z is (H/R)p. The required
moment is
k[
(q + 2
(^5) Show how the preceding problem gives the conclusion that the moment
of inertia of a homogeneous conical solid, having density 3 and height H and
base radius R, about a line through the vertex perpendicular to the axis is
-AVR2H(4H2 + R2) 3.
6 A solid cylinder having constant density 6 is bounded by the cylinder
and planes having the equations p = a, z = 0, and z = h. Set up and evaluate
a threefold iterated integral in cylindrical coordinates for the moment of inertia
I of the solid about the x axis. fins.:
I = s f 27 do f p dp h(z2 + p2 sin2 4,) dz = aa2h&[ 3z + 4
7 A conical solid has height h, base radius a, and density kz, where k is a
constant and z is the distance from the base. Find the mass M of the solid
and the distance f from the base to the centroid. Ins.: M = a2h2k, x = 32-h.
8 A conical solid has height h, base radius a, and density kp, where k is a
constant and p is the distance from the axis. Find the mass W of the solid and
the distance z from the base to the centroid. flns.: M =a3hk, I = 3h.
9 A cup-shaped solid S is obtained by rotation about the z axis of a region
R in the yz plane bounded by the graphs of the equations
2a ir
z=x2, z=x2+1, z=10.
The density (mass per unit volume) of S at the point having cylindrical coordi-