Calculus: Analytic Geometry and Calculus, with Vectors

4.5 Volumes and integrals 245

We now illustrate the "slab method" for finding volumes of three-
dimensional sets that are commonly called "solids." With the expecta-
tion that the method will be fully
understood and applied to find vol-
umes of othersolids, we find the vol-
ume of the solid coneof Figure 4.51
which consists of the points in E3
lying between the planes x = 0 and

x = h and inside or on the conical
surface. When we are not required

Figure 4.51

to explain the details of the method, we solve this problem in two lines

by writing

(4.52) Y = lim 11(x) Ox = foh d(x) dx

b \2 .b2 xa h

=

fhr

(hx l dx =

h2 3Jo =rb2h.

Even when we are not required to give explanations to someone else, we

do not write this without talking to ourselves. We make a partition P

of the interval 0 < x < h, but we draw only one subinterval having

length Ox and let x be a point of the subinterval. Planes perpendicular

to the x axis at the ends of the interval have between them a part of the

solid that we can call a slab. Let 14(x) be the area of the section in which

the solid intersects the plane which contains the point we have selected

and is perpendicular to the x axis. In case JPJ is small, the number

14(x) Ox is exactly equal to the volume of our slab or is a good approxi-

mation to it. We next write

(4.53) 214(x) Ax

and tell ourselves that this is either exactly or approximately the sum of

the volumes of the slabs and hence is exactly or approximately V. Hence

it should be true that

(4.54) Y = lim IJ(x) Ox.

But the right side of this formula is the integral in (4.52). Our next step

is to observe that 14(x) is the area of a circular disk whose radius y is

such that y/x = b/h and hence y = (bb/h)x. Thus

(4.55) 4(x) = 1r

()2

=rb x2.

With this information, we can quickly complete the work in (4.52).

Observe that it would not be easy to find the volume of the solid cone of

Figure 4.51 by employing slabs resulting from a partition of the interval

-b <_ y < b of the y axis. The difficulty resides in the fact that planes