Calculus: Analytic Geometry and Calculus, with Vectors

13.2 Iterated integrals and volumes 661

the parts covering Si, S2, S, have heights Z, B, C. We should know
that we can undertake to start with an x, y,z coordinate system oriented
in the usual way and to sketch a figure like Figure 13.23 showingthe
block (or hotel) H.

Figure 13.23 Figure 13.24

To find the volume of the block H, we do not need a figure in whichan
architect could take pride. It is sufficient to use the slab method which
was candidly presented and employed in Section 4.5. We make a parti-
tion P of the interval 0 < x < 2 into subintervals of lengths .x1, 0x2,
, Ax,,and let xk be a point in the kth subinterval so that xk_1
xk xk. For each k, the number

(13.241) fo'f(xk,y) dy

is the area of the intersection of the plane x = xk and the body H.
Depending upon the choice of xk, the number

(13.242) Dxk fo' f(xk,y) dy

is exactly equal to or is an approximation to the volume of the slab of
H between the planes x = xk_1 and x = xk. The sum in the formula

(13.243)

n

Y = lim Dxk f0' f(xk,y) dy

k=1

is then exactly equal to the volume P of H or is an approximation to V,
and in any case the limit as JP( --> 0 is V. Thus

(13.244) Y = I = fox dx f' f(x,y) dy