Calculus: Analytic Geometry and Calculus, with Vectors

682 Iterated and multiple integrals

Problems 13.49

1 A rectangular lamina has opposite vertices at the origin and the point

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

up an iterated integral for the pth moment of the lamina about the x axis. Then

evaluate the integral for the cases

(a) S(x,y) = 1 (b) S(x,y) = kx (c) S(x,y) = ky

Ans.: The required integral is

f oa dx fob YPS(x,Y) dy or fob dy fo YPS(x,Y) dx.

The required moments are respectively

abP+1 ka' kahp+2

p+ 1' 2(p -+I), p + 2

2 Supposing that 0 < p < q, set up and evaluate an iterated integral for

the area .4 of the region in the first quadrant bounded by the graphs of the

equations y = xP and y = x4. Ans.:

I xp q - p

fl- fo dxf.9 dy

(p + 1)(q + 1)

3 When 0 < p < q, the region in the first quadrant bounded by the graphs

of y = xP and y = x4 has area (q - p)/(p + 1)(q + 1). Find the coordinates

of the centroid of this region. 11ns.:

R (p+l)(q+1), Y (p+1)(g+l)

(p + 2)(q + 2) (2p + 1) (2q + 1)

4 Find the centroid of the region bounded by the x and y axes and the

graph of y= e=. fins.:x=1,y=TL

5 Find the centroid of the long golf tee obtained by rotating the region of

Problem 4 about the x axis. 4ns.: z = -ff, y' = 0, z = 0.

6 Find the centroid of the region in the strip 0 x r bounded by the x

axis and the graph of y = sin x. Ans.: x = yr/2, y = it/8.

7 Find the centroid of the region which lies in the interval 0 < x <= 2a

and is bounded by the graphs of the equations y = 0 and y = b sin Za As.:

a

a,8 b.

8 A vertical face of a dam is bounded by the segment 0 < x < 2a of the x

axis and the graph of the equation y = -b sin Za The water level is at the top

of the dam, and the weight per cubic unit of the water is w. Find the magnitude

of the force on the dam. As.: wab2.

(^9) Find the centroid of the region in the first quadrant bounded by the coordi-
nate axes and the hypocycloid having the equation x% + y% = a3. 4ns.:
256a
x=y=315ir'