Calculus: Analytic Geometry and Calculus, with Vectors

660 Iterated and multiple integrals

to existence of the integrals involved, we investigate the number I

defined by

(13.22) I = 0 2 dx f of f(x,y) dy.

Our first step is to recognize that, for each fixed x in the interval 0 <_

x < 2, the integrand in the integral

(13.221)

f1

f(x,y) dy

has values that depend upon y. To make effective use of our informa-

tion, we mark a point x on the x axis between 0 and 2 and then draw a

line through this point parallel to the y axis. A part of this line is in

Sl where f(x,y) = -I, another part is in S2 where f(x,y = B, a third

part is in S3 where f(x,y) = C, and, moreover, the end points of these

parts depend upon the fixed x. We must understand our situation so

thoroughly that we see that

f(x,y) = A when 0 < y < x2/4,

f (x,y) = B when x2/4 S y S x/2,

f(x,y) = C when x/2 < y < 1,

and that

(13.222)

so

(13.223)

f of f (x,Y) dy = fox

1

A dy + f x,/2 B dy + fz/2C dy

I f(x,y)dy=A + B (2!- 4)+C11-2

Substituting this in (13.22) gives

(13.224)

and hence

(13.225)

f2[x2(xx2)(

_)]dx

I =1A+*B+C.

Supposing that A, B, C are nonnegative constants, we proceed to

show that the number I is the volume of the solid block H which stands

upon the rectangular base R of Figure 13.21 and has, at each point

(x,y) of R, height f(x,y). It is quite possible to imagine that the rec-

tangular set R of Figure 13.21 lies in a horizontal plane beneath our eyes

and that the block H has its base on R and extends upward toward our

eyes. If the operation is helpful, we should imagine that we are in an

airplane and are looking down upon a hotel built upon the set or site R;