Calculus: Analytic Geometry and Calculus, with Vectors

13.2 Iterated integrals and volumes 665

find a region R such that the formula is valid whenever f(x,y) is continuous over
R and f(x,y) = 0 when the point (x,y) is not in R.
3 Find a region R such that the formula

f ol dx f s2if(x,y) dy= Iol dy f 12f(x,Y) dx + f2 dy L1121(x,Y) dx

is valid when f is continuous over R. Evaluate all of the integrals and make the

results agree when

(a) f(x,Y) = 1 (b) f(x,y) = x (c) f(x,Y) = Y

(d) f(x,y) = x + Y (e) f(x,y) = xy (f) f(x,Y) = x2 + y2

4 A particular solid body K can be described as the set in Ea which rests

upon the base S in the xy plane bounded by the plane graphs of the equations

y = x/2 and y = x2/4 and has, at each point (x,y) in S, height x2 + y2. The

same body K can be described as the set in Es which is bounded by the graphs

(they are all surfaces) in Ea of the equations

y=x/2, y=x2/4, z=0, z=x2+y2.

This book tries to be too honest to pretend that it is easy to sketch a good figure

showing the body K. The book does insist, however, that we should have picked

up ideas enough to enable us to use iterated integrals in two different ways to

find the volume IKI of K. Do it. Remark: The answers should agree with

each other. Moreover, since the area of the base is g and the height varies from

0 to 5, the answers should be between

0 and 1.

5 As in Figure 13.291, let S be the

closed set of points in the rectangle

a< x < b, c S Y S d which is bounded

below and above by the graphs of

y = fl(x) and y = f2(x) and which is

bounded on the left and right by graphs

of x = ga(y) and x = g2(y) Let F

be a function which is continuous

over S and is such that F(x,y) = 0

a b z

Figure 13.291

when (x,y) is a point not in S Tell why the first integral in the formula

(1) L d F(x,y) dy = ff (()) F(x,y) dy

exists and is equal to the second one when a S x < b. Tell why the first integral

in the formula

(2)

fab

F(x,y) dx = faM) F(x,y) dx

exists and is equal to the second one when c < y < d Show that the formula

(3) f b dx f dF(x,y) dY = f d dy f bF(xy) dxa c c a