Chapter 13.1, Problem 31E

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

Use rectangles to find the area between

*x*-axis from
*n* equal subintervals so that each subinterval has length 2/*n.*

To determine

**To calculate:** The area between the curve *x-*axis from *n* equal subintervals so that the length of each subinterval is

Explanation

**Given Information:**

The provided curve is *n* equal subintervals so that the length of each subinterval is

**Formula used:**

The base of the rectangles to approximate the area is *n* subintervals.

The height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

The area of a rectangle is

The approximated area under the curve is sum of the areas of each rectangle.

The value of the sum

The value of the sum

For functions

The value of

**Calculation:**

Consider the provided curve *n* equal subintervals so that the length of each subinterval is

The base of the rectangles to approximate the area is *n* subintervals.

Since, the function is defined from

Thus,

Thus, the *n* subintervals, each of length

Since, there are *n* subintervals, the number of rectangles is *n*.

Recall that the height of the rectangles is the value of the function calculated at the right-hand end point of the interval containing the base.

Since, the right-hand end point of the first interval

Thus, the height of the first rectangle is,

Recall that the area of a rectangle is

Thus, the area of the first rectangle is,

Do similar calculation to find the area of all *n* rectangles and record these values in a table.

Rectangle | Base | Right-endpoint | Height | |

Recall that the approximated area under the curve is sum of the areas of each rectangle.

Thus, the area under the curve is

Simplify

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