Chapter 13, Problem 11RCC

a.

To determine

To state: that how the approximated area can be find using rectangles.

Answer to Problem 11RCC

To find the approximated area using rectangles, first find the width and height of the rectangles and then multiply them.

Explanation of Solution

Given information:

The function is $y=f\left(x\right),a\le x\le b$.

to find the approximated area using rectangles, first find the width and height of the rectangles and then multiply them.

b.

To determine

To find: the expression of area as a limits of sums.

Answer to Problem 11RCC

The expression is $A=\underset{n\to \infty }{\lim }\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+f\left(x_3\right)\Delta x+\mathrm{...}+f\left(x_n\right)\Delta x\right]$

Explanation of Solution

Given information:

The function is $y=f\left(x\right),a\le x\le b$.

Calculation: the width of the rectangles,

  $\Delta x=\frac{b-a}{n}$

Then the area expression can be written as:

  $A=\underset{n\to \infty }{\lim }\left[f\left(x_1\right)\Delta x+f\left(x_2\right)\Delta x+f\left(x_3\right)\Delta x+\mathrm{...}+f\left(x_n\right)\Delta x\right]$