**Introduction to Volumes by Slicing**

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.

**Fig. 8.1**

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.

**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* .

**Fig. 8.2**

**Fig. 8.3**

If we can express the *area* of the slice at position *x* as a function *A* *(x*) of *x* , then (see Fig. 8.4) the *volume* of a slice of thickness Δ *x* at position *x* will be about *A* ( *x* ) · Δ *x*. If *P* = { *x* _{0} , *x* _{1} ,..., *x *_{k} } is a partition of the interval [ *a, b* ] then the volume of the original solid object will be about

**Fig. 8.4**

As the mesh of the partition becomes finer and finer, this (Riemann) sum will tend to the integral

We declare the value of this integral to be the *volume V of the solid object* .** **

**Volume of a Right Circular Cone with a Disc Cross-Section Example**

Calculate the volume of the right circular cone with base a disc of radius 3 and height 6.

**Solution**

Examine Fig. 8.5.

**Fig. 8.5**

We have laid the cone on its side, so that it extends from *x* = 0 to *x* = 6. The upper edge of the figure is the line *y* = 3 − *x* /2. At position *x*, the height of the upper edge is 3 − *x* /2, and that number is also the radius of the circular slice at position *x* (Fig. 8.6). Thus the area of that slice is

**Fig. 8.6**

We find then that the volume we seek is

**You Try It**: Any book of tables (see [CRC]) will tell you that the volume of a right circular cone of base radius *r* and height *h* is . This formula is consistent with the result that we obtained in the last example for *r* = 3 and *h* = 6. Use the technique of the example above to verify this more general formula.

**Volume of a Solid with an Equilateral Triangle Cross-Section Example**

A solid has base the unit disk in the *x-y* plane. The vertical cross section at position *x* is an equilateral triangle. Calculate the volume.

**Solution**

Examine Fig. 8.7.

**Fig. 8.7**

The unit circle has equation *x* ^{2} + *y* ^{2} = 1. For our purposes, this is more conveniently written as

Thus the endpoints of the base of the equilateral triangle at position *x* are the points . In other words, the base of this triangle is

Examine Fig. 8.8.

**Fig. 8.8**

We see that an equilateral triangle of side *b* has height . Thus the area of the triangle is . In our case then, the equilateral triangular slice at position *x* has area

Finally, we may conclude that the volume we seek is

**Volume of a Solid with a Square Cross-Section Example**

A solid has base in the *x-y* plane consisting of a unit square with center at the origin and vertices on the axes. The vertical cross-section at position *x* is itself a square. Refer to Fig. 8.9. What is the volume of this solid?

**Fig. 8.9**

**Solution**

It is sufficient to calculate the volume of the right half of this solid, and to double the answer. Of course the extent of *x* is then . At position *x*, the height of the upper edge of the square base is . So the base of the vertical square slice is ( Fig. 8.10 ). The area of the slice is then

**Fig. 8.10**

It follows that

**You Try It** : Calculate the volume of the solid with base in the plane an equilateral triangle of side 1, with base on the *x* -axis, and with vertical cross-section parallel to the *y* -axis consisting of an equilateral triangle.

**Volume Inside a Sphere with a Disk Cross-Section Example**

Calculate the volume inside a sphere of radius 1.

**Solution **

It is convenient for us to think of the sphere as centered at the origin in the *x-y* plane. Thus ( Fig. 8.11 ) the slice at position *x*, −1 ≤ *x* ≤ 1, is a disk. Since we are working with base the unit circle, we may calculate (just as in Example 8.2) that the diameter of this disk is . Thus the radius is and the area is

**Fig. 8.11**

In conclusion, the volume we seek is

We easily evaluate this integral as follows:

**You Try It**: Any book of tables (see [CRC]) will tell you that the volume inside a sphere of radius *r* is 4 *πr* ^{3} /3. This formula is consistent with the answer we obtained in the last example for *r* = 1. Use the method of this section to derive this more general formula for arbitrary *r*.

Find practice problems and solutions for these concepts at: Applications of the Integral Practice Test.

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From Calculus Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.