Page 217 - Schaum's Outline of Theory and Problems of Advanced Calculus

208 MULTIPLE INTEGRALS [CHAP. 9

where the limit is taken so that the number n of subdivisions increases without limit and such that the
largest linear dimension of each A k approaches zero. See Fig. 9-2(a). If this limit exists, it is denoted by

ð ð
Fðx; yÞ dA ð3Þ
r
and is called the double integral of Fðx; yÞ over the region r.
It can be proved that the limit does exist if Fðx; yÞ is continuous (or sectionally continuous) in r.
The double integral has a great variety of interpretations with any individual one dependent on the
form of the integrand. For example, if Fðx; yÞ¼ ðx; yÞ represents the variable density of a flat iron
Ð
plate then the double integral, dA,of this function over a same shaped plane region, A,is the mass of
A
the plate. In Fig. 9-2(b)we assume that Fðx; yÞ is a height function (established by a portion of a surface
z ¼ Fðx; yÞÞ for a cylindrically shaped object. In this case the double integral represents a volume.

Fig. 9-2

ITERATED INTEGRALS
If r is such that any lines parallel to the y-axis meet the boundary of r in at most two points (as is
true in Fig. 9-1), then we can write the equations of the curves ACB and ADB bounding r as y ¼ f 1 ðxÞ
and y ¼ f 2 ðxÞ, respectively, where f 1 ðxÞ and f 2 ðxÞ are single-valued and continuous in a @ x @ b.In this
case we can evaluate the double integral (3)by choosing the regions r k as rectangles formed by
constructing a grid of lines parallel to the x- and y-axes and A k as the corresponding areas. Then
(3) can be written

b f 2 ðxÞ
ðð ð ð
Fðx; yÞ dy dx
Fðx; yÞ dx dy ¼ ð4Þ
x¼a y¼f 1 ðxÞ
r
ð b ð f 2 ðxÞ
Fðx; yÞ dy dx
¼
x¼a y¼f 1 ðxÞ