A L T A I R

Areas and Volumes

Theorem 8.1.1.  The area \( A \) of the region \( S \) between the graph of the continuous function \( f \) and the \( x \)-axis is

\( \displaystyle A = \int_{a}^{b} \left|\ f(x)\ \right| \operatorname{d}\!x \)

where \( a \le x \le b \).

Theorem 8.1.2.  The area between the curves \( y = f(x) \) and \( y = g(x) \) and between \( x = a \) and \( x = b \), where \( a \le x \le b \), is

\( \displaystyle A = \int_{a}^{b} \left|\ f(x) - g(x)\ \right| \operatorname{d}\!x \)

Figure 1

Proof.  Consider the region \( S \) that lies between two curves \( y = f(x) \) and \( y = g(x) \) and between the vertical lines \( x = a \) and \( x = b \), where \( f(x) \ge g(x) \) for all \( x \) in \( \left[a,\ b\right] \).

Just as we did in Definition 7.2.1, we divide \( S \) into \( n \) strips of equal width and then we approximate the \( i \)th strip by a rectangle with base \( \Delta x \) and height \( f(x_{i}^{*}) - g(x_{i}^{*}) \). The Riemann sum

\( \displaystyle \sum_{i = 1}^{n}\left[ f(x_{i}^{*}) - g(x_{i}^{*}) \right] \Delta x \)

is therefore an approximation of the area of \( S \).

This approximation appears to become better and better as \( n \to \infty \). Therefore the area of the region \( S \) is the limiting value of the sum of the areas of these approximating rectangles.

\( \displaystyle A = \lim_{n \to \infty}\sum_{i = 1}^{n}\left[ f(x_{i}^{*}) - g(x_{i}^{*}) \right] \Delta x \)

And by Definition 7.2.1,

\( \displaystyle \begin{align} A &= \lim_{n \to \infty}\sum_{i = 1}^{n}\left[ f(x_{i}^{*}) - g(x_{i}^{*}) \right] \Delta x \\[7pt] &= \int_{a}^{b}\left[f(x) - g(x)\right] \operatorname{d}\!x \qquad \cdots \qquad \text{(i)} \end{align} \)

If \( g(x) \ge f(x) \), since heights must be positive, the height of the rectangles is \( g(x_{i}^{*}) - f(x_{i}^{*}) \). Therefore

\( \displaystyle \begin{align} A &= \lim_{n \to \infty}\sum_{i = 1}^{n}\left[ g(x_{i}^{*}) - f(x_{i}^{*}) \right] \Delta x \\[7pt] &= \int_{a}^{b}\left[g(x) - f(x)\right] \operatorname{d}\!x \qquad \cdots \qquad \text{(ii)} \end{align} \)

By \( \text{(i)} \) and \( \text{(ii)} \), the area \( A \) is

\( \displaystyle A = \int_{a}^{b}\left|\ f(x) - g(x)\ \right| \operatorname{d}\!x \)

\( \Box \)

Example 8.1.3.  Find the area of the region bounded by the curves \( y = \sin x \), \( y = \cos x \), \( x = 0 \), and \( x = \pi/2 \).

Solution.  The points of intersection occur when \( \sin x = \cos x \), that is, when \( x = \pi/4 \) (since \( 0 \le x \le \pi/2 \)). Notice that \( \cos x \ge \sin x \) when \( 0 \le x \le \pi/4 \) but \( \sin x \ge \cos x \) when \( \pi/4 \le x \le \pi/2 \). Therefore the required area is

\( \displaystyle \begin{align} A &= \int_{0}^{\large \frac{\pi}{2}} \left|\ \cos x - \sin x\ \right| \operatorname{d}\!x \\[7pt] &= \int_{0}^{\large \frac{\pi}{4}}(\cos x - \sin x) \operatorname{d}\!x + \int_{\large \frac{\pi}{4}}^{\large \frac{\pi}{2}}(\sin x - \cos x) \operatorname{d}\!x \\[7pt] &= \Big[ \sin x + \cos x \Big]_{0}^{\pi/4} + \Big[ -\cos x - \sin x \Big]_{\pi/4}^{\pi/2} \\[7pt] &= \left( \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 0 - 1 \right) + \left( - 0 - 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \right) \\[7pt] &= 2\sqrt{2} - 2 \end{align} \)

\( \Box \)

Theorem 8.1.4.  Let \( S \) be a solid that lies between \( x = a \) and \( x = b \) (\( a \le x \le b \)). If the cross-sectional area of \( S \) in the plane \( P_{x} \), through \( x \) and perpendicular to the \( x \)-axis, is \( A(x) \), where \( A \) is a continuous function, then the volume of \( S \) is

\( \displaystyle V = \lim_{n \to \infty} \sum_{i = 1}^{n} A(x_{i}^{*}) \Delta x = \int_{a}^{b}A(x) \operatorname{d}\!x \)

Figure 2

Figure 3

Theorem 8.1.5.  The volume of the solid, obtained by rotating about the \( x \)-axis the region between the curve \( y = f(x) \) and the \( x \)-axis from \( a \) to \( b \), is

\( \displaystyle V = \int_{a}^{b}\pi\left[f(x)\right]^{2} \operatorname{d}\!x \)

Also, the volume of the solid obtained by rotating about the \( y \)-axis the region between the curve \( y = f(x) \) and the \( x \)-axis from \( a \) to \( b \), is

\( \displaystyle V = \int_{a}^{b}2\pi \left|\ x\ \right|f(x) \operatorname{d}\!x \)

The body text and the figures were excerpted from James Stewart, Calculus 8E.