Page 230 - Calculus Demystified

CHAPTER 8

Applications of

the Integral

8.1 Volumes by Slicing

8.1.0 INTRODUCTION

When we learned the theory of the integral, we found that the basic idea was that
one can calculate the area of an irregularly shaped region by subdividing the region
into “rectangles.” We put the word “rectangle” here in quotation marks because the
region is not literally broken up into rectangles; the union of the rectangles differs
from the actual region under consideration by some small errors (see Fig. 8.1). But
the contribution made by these errors vanishes as the mesh of the rectangles become
finer and finer.
We will now implement this same philosophy to calculate certain volumes. Some
of these will be volumes that you have heard about (e.g., the sphere or cone), but
have never known why the volume had the value that it had. Others will be entirely
new (e.g., the paraboloid of revolution). We will again use the method of slicing.

8.1.1 THE BASIC STRATEGY

Imagine a solid object situated as in Fig. 8.2. Observe the axes in the diagram, and
imagine that we slice the figure with slices that are vertical (i.e., that rise out of
the x-y plane) and that are perpendicular to the x-axis (and parallel to the y-axis).
Look at Fig. 8.3. Notice, in the figure, that the figure extends from x = a to x = b.

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