Chapter 13.5, Problem 2E

To determine

To Fill: The exact area under the graph of $f$.

Answer to Problem 2E

The exact area under the graph of $f$ is $\underset{n\to \infty }{\lim }{\displaystyle \sum _{r=1}^{n}f\left(x_r\right)}\cdot \Delta x$.

Explanation of Solution

To calculate areas under the graph divide the area by using $n$ rectangle of equal width.

To achieve accuracy increase the number of rectangle.

Number of rectangle increases, error in calculation of area decreases.

$R_n$ is the sum of areas of n rectangles with width $\frac{1}{n}$ and heights are the values of the function $f\left(x\right)$.

$R_n=f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+f\left(x_3\right)\Delta x+\cdot \cdot \cdot f\left(x_n\right)\Delta x$ (1)

Where, $f\left(x_1\right)$ is height of the rectangle, $\Delta x$ is width of the rectangle.

Use summation and rewrite equation (1) as follows,

$R_n={\displaystyle \sum _{r=1}^{n}f\left(x_r\right)}\Delta x$

To get exact area take limit of above equation with $n\to \infty$

$\text{A}=\underset{n\to \infty }{\lim }{\displaystyle \sum _{r=1}^{n}f\left(x_r\right)\cdot }\Delta x$

Thus, the value of exact area under the function $f$ is $\underset{n\to \infty }{\lim }{\displaystyle \sum _{r=1}^{n}f\left(x_r\right)\cdot }\Delta x$.