Chapter 11.4, Problem 55E

Solution

To devise a limit-of-sums definition of area of the region between the two curves over the interval $\left[a,b\right]$ .

To find the area between the two curves $f\left(x\right)$ and $g\left(x\right)$ over the interval $\left[a,b\right]$ , the interval can be divided into $n$ -subintervals of equal length. This will divide the whole area into small rectangles each of width $\Delta x=\frac{b-a}{n}$ . Then, left end points of each interval will be $x_i=a+i\frac{b-a}{n}$ .

Therefore, the area of each rectangle at $x_i=a+i\frac{b-a}{n}$ is $\left(f\left(x_i\right)-g\left(x_i\right)\right)\Delta x$ .

So, the area between the curves is given by $A={\displaystyle \sum _{i=0}^n\left(f\left(x_i\right)-g\left(x_i\right)\right)\Delta x}$ .

Thus, the limit-of-sum for the area between the curve can be given as $A=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=0}^n\left(f\left(x_i\right)-g\left(x_i\right)\right)\Delta x}$ or $A=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=0}^n\left(\left(f\left(a+i\frac{b-a}{n}\right)-g\left(a+i\frac{b-a}{n}\right)\right)\frac{b-a}{n}\right)}$ .

If the area under both the curves are already known, then area under the below curve can be subtracted from the area under the upper curve to get the area between the two curves.