Volume and the Slicing Method

Just as area is the numerical measure of a two-dimensional region, volume is the numerical measure of a three-dimensional solid. Most of us have computed volumes of solids by using basic geometric formulas. The volume of a rectangular solid, for example, can be computed by multiplying length, width, and height: \(V = l w h .\) The formulas for the volume of a sphere \(\left(\right. V = \frac{4}{3} \pi r^{3} \left.\right) ,\) a cone \(\left(\right. V = \frac{1}{3} \pi r^{2} h \left.\right) ,\) and a pyramid \(\left(\right. V = \frac{1}{3} A h \left.\right)\) have also been introduced. Although some of these formulas were derived using geometry alone, all these formulas can be obtained by using integration.

We can also calculate the volume of a cylinder. Although most of us think of a cylinder as having a circular base, such as a soup can or a metal rod, in mathematics the word cylinder has a more general meaning. To discuss cylinders in this more general context, we first need to define some vocabulary.

We define the cross-section of a solid to be the intersection of a plane with the solid. A cylinder is defined as any solid that can be generated by translating a plane region along a line perpendicular to the region, called the axis of the cylinder. Thus, all cross-sections perpendicular to the axis of a cylinder are identical. The solid shown in Figure 6.11 is an example of a cylinder with a noncircular base. To calculate the volume of a cylinder, then, we simply multiply the area of the cross-section by the height of the cylinder: \(V = A \cdot h .\) In the case of a right circular cylinder (soup can), this becomes \(V = \pi r^{2} h .\)

This graphic has two figures. The first figure is half of a cylinder, on the flat portion. The cylinder has a line through the center labeled “x”. Vertically cutting through the cylinder, perpendicular to the line is a plane. The second figure is a two dimensional cross section of the cylinder intersecting with the plane. It is a semi-circle.

Figure 6.11 Each cross-section of a particular cylinder is identical to the others.

If a solid does not have a constant cross-section (and it is not one of the other basic solids), we may not have a formula for its volume. In this case, we can use a definite integral to calculate the volume of the solid. We do this by slicing the solid into pieces, estimating the volume of each slice, and then adding those estimated volumes together. The slices should all be parallel to one another, and when we put all the slices together, we should get the whole solid. Consider, for example, the solid S shown in Figure 6.12, extending along the \(x -\text{axis} .\)

This figure is a graph of a 3-dimensional solid. It has one edge along the x-axis. The x-axis is part of the 2-dimensional coordinate system with the y-axis labeled. The edge of the solid along the x-axis starts at a point labeled “a” and stops at a point labeled “b”.

Figure 6.12 A solid with a varying cross-section.

We want to divide \(S\) into slices perpendicular to the \(x -\text{axis} .\) As we see later in the chapter, there may be times when we want to slice the solid in some other direction—say, with slices perpendicular to the y-axis. The decision of which way to slice the solid is very important. If we make the wrong choice, the computations can get quite messy. Later in the chapter, we examine some of these situations in detail and look at how to decide which way to slice the solid. For the purposes of this section, however, we use slices perpendicular to the \(x -\text{axis} .\)

Because the cross-sectional area is not constant, we let \(A \left(\right. x \left.\right)\) represent the area of the cross-section at point \(x .\) Now let \(P = \left{\right. x_{0} , x_{1} \ldots , X_{n} \left.\right}\) be a regular partition of \(\left[\right. a , b \left]\right. ,\) and for \(i = 1 , 2 ,\ldots n ,\) let \(S_{i}\) represent the slice of \(S\) stretching from \(x_{i - 1} \text{to} x_{i} .\) The following figure shows the sliced solid with \(n = 3 .\)

Figure 6.13 The solid \(S\) has been divided into three slices perpendicular to the \(x -\text{axis} .\)

Finally, for \(i = 1 , 2 ,\ldots n ,\) let \(x_{i}^{\star}\) be an arbitrary point in \(\left[\right. x_{i - 1} , x_{i} \left]\right. .\) Then the volume of slice \(S_{i}\) can be estimated by \(V \left(\right. S_{i} \left.\right) \approx A \left(\right. x_{i}^{\star} \left.\right) \Delta x .\) Adding these approximations together, we see the volume of the entire solid \(S\) can be approximated by

\[V \left(\right. S \left.\right) \approx \sum_{i = 1}^{n} A \left(\right. x_{i}^{\star} \left.\right) \Delta x .\]

By now, we can recognize this as a Riemann sum, and our next step is to take the limit as \(n \rightarrow \infty .\) Then we have

\[V \left(\right. S \left.\right) = \underset{n \rightarrow \infty}{\text{lim}} \sum_{i = 1}^{n} A \left(\right. x_{i}^{\star} \left.\right) \Delta x = \int_{a}^{b} A \left(\right. x \left.\right) d x .\]

The technique we have just described is called the slicing method. To apply it, we use the following strategy.

Problem-Solving Strategy

Finding Volumes by the Slicing Method

Examine the solid and determine the shape of a cross-section of the solid. It is often helpful to draw a picture if one is not provided.
Determine a formula for the area of the cross-section.
Integrate the area formula over the appropriate interval to get the volume.

Recall that in this section, we assume the slices are perpendicular to the \(x -\text{axis} .\) Therefore, the area formula is in terms of x and the limits of integration lie on the \(x -\text{axis} .\) However, the problem-solving strategy shown here is valid regardless of how we choose to slice the solid.

Example 6.6

Deriving the Formula for the Volume of a Pyramid

We know from geometry that the formula for the volume of a pyramid is \(V = \frac{1}{3} A h .\) If the pyramid has a square base, this becomes \(V = \frac{1}{3} a^{2} h ,\) where \(a\) denotes the length of one side of the base. We are going to use the slicing method to derive this formula.

Solution

We want to apply the slicing method to a pyramid with a square base. To set up the integral, consider the pyramid shown in Figure 6.14, oriented along the \(x -\text{axis} .\)

This figure has two graphs. The first graph, labeled “a”, is a pyramid on its side. The x-axis goes through the middle of the pyramid. The point of the top of the pyramid is at the origin of the x y coordinate system. The base of the pyramid is shaded and labeled “a”. Inside of the pyramid is a shaded rectangle labeled “s”. The distance from the y-axis to the base of the pyramid is labeled “h”. the distance the rectangle inside of the pyramid to the y-axis is labeled “x”. The second figure is a cross section of the pyramid with the x and y axes labeled. The cross section is a triangle with one side labeled “a”, perpendicular to the x-axis. The distance a is from the y-axis is h. There is another perpendicular line to the x-axis inside of the triangle. It is labeled “s”. It is x units from the y-axis.

Figure 6.14 (a) A pyramid with a square base is oriented along the x-axis. (b) A two-dimensional view of the pyramid is seen from the side.

We first want to determine the shape of a cross-section of the pyramid. We know the base is a square, so the cross-sections are squares as well (step 1). Now we want to determine a formula for the area of one of these cross-sectional squares. Looking at Figure 6.14(b), and using a proportion, since these are similar triangles, we have

\[\frac{s}{a} = \frac{x}{h} \text{or} s = \frac{a x}{h} .\]

Therefore, the area of one of the cross-sectional squares is

\[A \left(\right. x \left.\right) = s^{2} = \left(\right. \frac{a x}{h} \left.\right)^{2} \left(\right. \text{step} 2 \left.\right) .\]

Then we find the volume of the pyramid by integrating from \(0 \text{to} h\) (step \(3 \left.\right) :\)

\[\begin{aligned} V & = \int_{0}^{h} A \left(\right. x \left.\right) d x \\ & = \int_{0}^{h} \left(\right. \frac{a x}{h} \left.\right)^{2} d x = \frac{a^{2}}{h^{2}} \int_{0}^{h} x^{2} d x \\ & = \left(\left[\right. \frac{a^{2}}{h^{2}} \left(\right. \frac{1}{3} x^{3} \left.\right) \left]\right. \left|\right.\right)_{0}^{h} = \frac{1}{3} a^{2} h . \end{aligned}\]

This is the formula we were looking for.

Checkpoint 6.6

Use the slicing method to derive the formula \(V = \frac{1}{3} \pi r^{2} h\) for the volume of a circular cone.