Calculus: Analytic Geometry and Calculus, with Vectors

13.2 Iterated integrals and volumes 659

Ans.: In each case the answer is u(x,y), and itmay be worthwhile to try to under-

stand why this should be so.

18 Try to understand the formulas

Ibv uv(x,t) dt = u(x,t) It

J_V

mb= u(x,y) - u(x,b)

and

x J

a ds b

v

zt,,v(s>t) dt xds zt s t

teb

[us(s,y)-u,(s,b)] ds=[u(s,1)-u(s,b)]

a aaa

= u(x,y) - u(a,y) - u(x,b) + u(a,b).

13.2 Iterated integrals and volumes

multiple integrals and their applications, we continually need unin-

herited skills and information that can be efficientlyacquired by making

a calm and thorough examination of matters relating to Figure 13.21.

Figure 13.21

We start with the idea that the graphs of the two equationsy = x/2 and
y = x2/4 intersect at the points (0,0) and (2,1). These graphs separate
the closed rectangular region R, consisting of points (x,y) for which
0 < x S 2, 0 =< y < 1, into three subsets S1, S2, S3. While it makes no
difference how disputes over ownership of boundaries are resolved, we
want them resolved in some way and we suppose that S2 contains each
of its boundary points. Thus S2 is the set of points (x,y) for which 0
x 5 2 and x2/4 < y S x/2. Then S1 is the set of points (x,y) for which
0 < x <_ 2 and 0 < y < x2/4, and S3 is the set of points (x,y) for which
0-<-x<2andx/2<y<=1.

While more recondite modifications of the construction are easily made,
we keep our example simple by supposing that A, B, C are three constants,
that f(x,y) = A when (x,y) is a point of Si, that f(x,y) = B when (x,y)
is a point of S2, and that f(x,y) = C when (x,y) is a point of S3. Subject