Chapter 11.5, Problem 42E

To determine

To identify true or false:

Answer to Problem 42E

The exact area of a region is given by the limit of the sum of $n$ rectangles as $n$ approaches $0$ is false.

Explanation of Solution

Given Statement:

The exact area of a region is given by the limit of the sum of $n$ rectangles as $n$ approaches $0$ .

Consider the following statement,

The exact area of a region is given by the limit of the sum of $n$ rectangles as $n$ approaches $0$ .

The exact area of a region is given by formula,

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

Therefore, the exact area of a region is given by the limit of the sum of $n$ rectangles as $n$ approaches $\infty$ .

Hence the above statement is false.