Calculus: Analytic Geometry and Calculus, with Vectors

666

reduces to

(4)

Iterated and multiple integrals

f

b

dx

f (z) d Y)

f

F(x,y) dy = f dy

pz

f

F(x,y) dx.

a !3(z) c 0i(Y)

Finally, give geometric interpretations of the numbers in (4) when, for each point

(x,y) in S,

(a) F(x,y) = 1

(b) F(x,y) = xz + yz

6 By use of iterated integrals, find the volume Y of the solid bounded by

the three coordinate planes and the graph of the equation

x y z

Remark and ans.: The solid is a pyramid, and use of the fundamental fact that

the volume of a pyramid is one-third of the product of its height and the area of

its base gives F = wabc

7 Even when details are efficiently managed, it is not a short task to find

the volume V of the solid in the first octant bounded by the three coordinate

planes and the graph of the equation

\a/;F+\b)"+

It is, however, worthwhile to try to manage details efficiently and to earn the

satisfaction involved in showing that F = abc/90. It can be added that the

world is wide enough to accommodate and even need persons who run amok or

amuck and become strong enough to solve the problems obtained by replacing the

exponent by s and ' and

8 Prove that if the formula

(1) f ab dx f'(}(x,y) + B) dy = f d dy f ab [}(x,9') + B] dx

is valid for some constant B, then

(2)

fab

dx fdf(x',y) dy= 1 d dyf I f(x,y) dx.

Remark: In case f is a bounded function for which f(x,y) is sometimes positive

and sometimes negative, we can choose B such that f(x,y) + B is always positive.

In appropriate circumstances, we can prove (2) by proving the first formula

which involves only positive integrands.

9 Let a and x be confined to an interval over which a given function f is

continuous. Let fo(x) = f(x) and let

(1) fi(x') = fazf(t) dt, fz(x') = fazfl(t) dt, fa(x) = fazfz(t) dt

and so on so that, for each n = 1, 2, 3, ,

(2) f. (x) = fazfn(t) A