Calculus: Analytic Geometry and Calculus, with Vectors

13.4 Applications of double and iteratedintegrals 683

10 Solve Problem 9 again, using the parametric equations

to obtain integrals involving i.
11 A lamina having density xy at the point P(x,y) lies in the first quadrant
and is bounded by the coordinate axes and the ellipse having the equation

xz^2

a2-+ -bz = 1.

Find its mass M and the coordinates of its centroid. An:.: .41 = a2b2 z =

a

A., y = -rsb.

12 For the region R bounded by the positive x axis and the graphs ofr =

xe : and x = a, find each at the following:

(a) The area of R Ans.: 1 - (a + 1)e a

(b) The volume of the solid obtained by rotating .R about the x axis

Ans.: -,r[1 - (1 + 2a + 2a2)e Za]

(c) The volume of the solid obtained by rotating R about the y axis

Ans.: 2a[2 - (2 + 2a + a2)e a]

(d) The first moment of R about the x axis Ans.: $[1 - (1 + 2a + 2a2)e26]

(e) The first moment of R about the y axis Ans.: 2 - (2 + 2a + a2)e

(f) The moment of inertia of R about the x axis

Ans.: r",r[2 - (2 + 6a + 9a2 + 9aa)e3"]

(g) The moment of inertia of R about the y axis

Ans.: 6 - (6 + 6a + 3a2 + aa)e°

(h) The polar moment of inertia of R about the line through the origin perpen-

dicular to the plane of R Ans.: Sum of answers to (f) and (g)

13 A triangular lamina has vertices at points PI(x1,yi), Pz(x2,y2), P3(xa,ya)

and has areal density (mass per unit area) S(x,y) at the point (x,y). Assuming

that the points are placed as in Figure 13.491 so that x1 < xa < x2 and y2 < yi <

Y

x = a cosy t, y = a sin3 t

P

0 xo x1 xa x2 x

Figure 13.491

ya, and letting ink denote the slope of the side oppositePk so that

m1 =ys - y M2 =ya - Yi ma =Y2-Yi,

xa - x2 xa - x1 x2 - x1

set up an iterated integral for the pth momentMsp>ZO of the lamina about the