Calculus: Analytic Geometry and Calculus, with Vectors

13.3 Double integrals 667

Since 0! = 1, the formula

(3) M-) r x (x

= a (n - 1)! f(t)dt

is certainly correct when n = 1. Assuming that (3) is correct fora particular

positive integer n, show that

(4) fn (u) = f.u (n(n- 1) f(t) dt

and use (2) to obtain

(un--t)n

1)

I

(5) f+,+l(x) = fax du Lu( f(t) dt.

Use (5) and Figure 13.292 to obtain

(6) f

x x

Figure 13.292

and then use (6) to obtain the result of replacingn by n + 1 in (3). Since (3)

is correct when n = 1, it must be correct whenit = 2 and hence when it = 3 and

hence when it. = 4, and so on.

(^10) This section should not leave the impression thatour ideas about Riemann
integrals can always be applied to Riemann-Cauchy integrals. It can happen
that
I = f. dx f. f(x,y) dy, j = f dy f f(x,y) dx
both exist but have unequal values. For example, let
f (x,y) = 1 when x >0, x - 1 <y<x,

f(x,y) = -1when x>0,x <y <x+1,

f (x,y) = 0 otherwise.

Show that, in this case, I = 0 and J = 1.

13.3 Double integrals Section 4.2 showed how we partition intervals

into subintervals to form Riemann sums and how we use these sums to
define Riemann integrals over one-dimensional intervals. Because the
idea is important in both pure and applied mathematics, we must learn
about the process by which a set S in a plane is partitioned into subsets
in order to enable us to form Riemann sums and define Riemann integrals
over S. To begin, let S be a set of points which may, for example, be
the set of points inside and on a circle or an outer boundary curve such as
that shown in Figure 13.31. The set S may be the set of points P which
are neither outside the outer boundary nor inside the inner boundary of
Figure 13.32. What we really require is that the set S have positive
area and that the points of S lie inside or on some rectangle R so that the
set S is bounded. In some applications the set S is regarded as a lamina
(thin plate) or as a plane section of a three-dimensional solid in which
we are interested. While we normally use coordinates (rectangular or