Chapter 13, Problem 4RE

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

Use rectangles to find the exact area under the graph of
*x* = 0 to *x* = 1. Use *n* equal subintervals.

To determine

**To calculate:** The area under the graph of *n* equal-intervals of the same size and use rectangles to find the exact area.

Explanation

**Given Information:**

The provided function 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 the sum of the areas of each rectangle.

The value of the sum

**Calculation:**

The curve is *n* subintervals and the function is evaluated at right-hand end points of the subintervals.

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 subinterval

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 a 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

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