Calculus: Analytic Geometry and Calculus, with Vectors

13.5 Integrals in polar coordinates 691

gives the approximation

(13.58) S(P,0)P3 AP 0O

to the polar moment of the subset. This leads to the formulas

MZZo v_Vu = lim ES(pAP, AP 0o

and

M<2> a f(o)

X=o vow = d f0 6(P,o)P3 dp.

Problems 13.59

(^1) Set up an iterated integral in polar coordinates for the volume 1' of the
solid generated by rotating, about the y axis, the triangular set bounded by the
lines having the polar equations 0 = 0 and 0 = 0 (where 0 < S < 7r'2) and
the line having the rectangular equation x = k (where h > 0). Then evaluate
the integral and discover that Y = 23 ha tan 0 Remark: Correctness of the
answer can be verified by use of elementary geometry, because the solid is
obtainable by removing a part of a solid right circular cone from a segment of a
solid right circular cylinder.
(^2) Find the distance from the vertex to the centroid of a lamina having the
foim of a circular sector of radius R and central angle 2a when each of the fol-
lowing is true.
(a) The lamina is uniform. lgns : 3 siaa R
(b) The density is proportional to kth power of the distance from the vertex.
11ns.:k+2sinaRk+3 a
3 Using the equation p = 2a cos ¢, set up and evaluate an iterated integral
in polar coordinates for the moment of inertia to of a circular disk about a line
perpendicular to the disk and containing a point on the boundary of the disk.
.4ns..
Ia= 2fr x/2v/2 2a cos ¢
d4 f0 p0dp=
a4.
4 Using the equation p = a, set up an iterated integral in polar coordinates
for the moment of inertia Ia of a circular disk about the axis of the disk (the line
through the center of the disk perpendicular to the plane of the disk) when the
density of the disk at each point is the pth power of the distance from the point to
the diameter on the initial line (or x axis). 14ns.:
lo = 4 fo7/2 sin'o do foa p'+= dp.
Remark: With the aid of the formula
p-1 (q-1
fog/2
sin' x cosq x dx = 2
2
C+ 42 2
1 1