Calculus: Analytic Geometry and Calculus, with Vectors

664 Iterated and multiple integrals

a set S which is a subset of the rectangular set R containing points (x,y)

for which a < x < b, c < y < d, the height of K at each point (x,y)

in S being f(x,y). We can undertake to solve the problem in the fol-

lowing way. Let f be extended in such a way that f(x,y) = 0 when

(x,y) is a point not in the set S. Then, unless the set S and the function

f are much more tortuous than those appearing in elementary nonpatho-

logical problems, two applications of the slab method for finding the

volume of V yield the formula

(13.28) Y =

fb

dx f df(x,y) dY = f

a

dy f bf(x,y) dx.

It is very important to be aware that, in all ordinary and many extra-

ordinary circumstances, the last two members of (13.28) are equal even

when f is a discontinuous function for which f(x,y) = 0 when the point

(x,y) is not in a particular set S in which we are interested. More

information about this matter will appear later.

Problems 13.29

1 Let

I

f 4o

dxf_40f (x,y) dy, J = f 440 dv.l440 f(x,y) dx,

where f is continuous over the region R bounded by the graphs of the lines having

the equations y = -1, y = x, and x = 1 and f(x,y) = 0 when the point (x,y)

is not in R. Show that

I = fit dx f 21 f(x,y) dy, j = f 11 dy f vl f(x,y) dx.

Evaluate I and J and show that they are equal in case

(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

when (x,y) is in R and f(x,y) = 0 when (x,y) is not in R.

2 For each of the formulas

(a) f dx f f(x,y) dy = f ' dx fox f(x,y) dy

(b) f dy f f(x,y)dx=104 dy 'f(x,y)dx

(e) f d x f f(x,y) dy = fo4 dx f

xf(x,y)

dy

( d ) foa dx fo°f(x,y) d y = foa dx

f0

a=-x

f(x,Y) dy