Chapter 12.5, Problem 47E

To determine

To describe: The process of finding the area of a region bounded by the graph of a nonnegative, continuous function $f$ .

Answer to Problem 47E

Approximate area $A\approx {\displaystyle \sum _i^{n}f}\left(a+\frac{\left(b-a\right)}{n}i\right)\frac{b-a}{n}$ .

Exact area $A=\underset{n\to 0}{\lim }{\displaystyle \sum _{i=1}^{n}f}\left(a+\frac{\left(b-a\right)}{n}i\right)\frac{b-a}{n}$ .

Explanation of Solution

Given information:

The area of a region is bounded by the graph of a nonnegative continuous function.

  $\begin{array}{l}x=a\\ x=b\end{array}$

Calculation:

From the given,

The graph of the function bounded by the vertical lines and $x-\text{axis}$ .

Width of the rectangle is $\frac{b-a}{n}$ .

Height of the rectangle is $f\left(a+\frac{\left(b-a\right)}{n}\right)i$

Therefore the approximate area is $A\approx {\displaystyle \sum _i^{n}f}\left(a+\frac{\left(b-a\right)}{n}i\right)\frac{b-a}{n}$ .

To find the accurate area, take the limit as $n\to 0$ ,

  $A=\underset{n\to 0}{\lim }{\displaystyle \sum _{i=1}^{n}f}\left(a+\frac{\left(b-a\right)}{n}i\right)\frac{b-a}{n}$

Thus we can use the limit process to the area of the region bounded.