Chapter 6, Problem 1PE

To determine

To describe: The process for finding the area between two curves.

Explanation of Solution

The process for finding the area between two curves:

Suppose $f$ and $g$ are functions such that $f(x)\ge g(x)$ on the interval $[a,b]$ , as shown in Figure 6.1.

To find the area of the region $R$ between the curves defined by $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$ , subtract the area between the lower curve $y=g(x)$ and the x-axis from the area between the upper curve $y=f(x)$ and the x-axis.

That is,

  $\begin{array}{l}\text{Area of }R={\displaystyle \underset{a}{\overset{b}{\int }}f(x)dx}-{\displaystyle \underset{a}{\overset{b}{\int }}g(x)dx}\\ ={\displaystyle \underset{a}{\overset{b}{\int }}\left[f(x)-g(x)\right]dx}\end{array}$

Suppose $f$ and $g$ are continuous such that $f(x)\ge g(x)$ on the interval $[a,b]$ .

To find the area of the region between the curves defined by $y=f(x)$ and $y=g(x)$ from $x=a$ to $x=b$ , choose a partition

  $[x_0=a,x_1,x_2,\mathrm{...},x_n=b]$

of the interval $\left[a,b\right]$ and a representative number $x_k^{+}$ from each subinterval $\left[x_{k-1},x_k\right]$ .

For each index $k$ , with $k=1,2,\mathrm{...},n$ , construct a rectangle of width,

  $\Delta x_k=x_k-x_{k-1}$

and height,

  $f(x_k^{+})-g(x_k^{+})$

equal to the vertical distance between the two curves at $x=x_k^{+}$ .

Refer to the approximating rectangle as a vertical strip which is shown in Figure 6.3.

The representative rectangle has area

  $\Delta A_k=\left[f(x_k^{+})-g(x_k^{+})\right]\Delta x_k$

And the total area between the curves $y=f(x)$ and $y=g(x)$ can be estimated by the Riemann sum

  $A_n={\displaystyle \sum _{k=1}^n\left[f(x_k^{+})-g(x_k^{+})\right]\Delta x_k}$ .

Increase the number of subdivision points in the partition $P$ in such a way that the norm $\Vert P\Vert$ approaches $0$ .

Thus, the region between the two curves has area

  $A=\underset{\Vert P\Vert \to 0}{\lim }{\displaystyle \sum _{k=1}^n\left[f(x_k^{+})-g(x_k^{+})\right]\Delta x_k}$

This is the integral of the function $f(x)-g(x)$ over the interval $[a,b]$ .

Using all the above observations, the area between the two curves can be defined as,

If $f$ and $g$ are continuous and satisfy $f(x)\ge g(x)$ on the closed interval $[a,b]$ , then the area between the two curves $y=f(x)$ and $y=g(x)$ is given by

  $A={\displaystyle \underset{a}{\overset{b}{\int }}\left[f(x)-g(x)\right]dx}$ .

Conclusion: Thus, the area between two curves is $A={\displaystyle \underset{a}{\overset{b}{\int }}\left[f(x)-g(x)\right]dx}$ .