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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 5.1, Problem 1E

**(a)**

To determine

The lower estimate and upper estimate ofthe area under a graph using five rectangles.

Expert Solution

The lower estimate of the area under the graph is 12.

The upper estimate of the area under the graph is 22.

**Given information:**

Take the curve as

Consider that the region lies between

Consider the number of rectangles as

Use the right end points of the curve as the lower estimate and the left end points of the curve as the upper estimate for decreasing behavior of the curve (*f*).

The expression to find the lower estimate of the areas of *n* rectangles

Here, the height of first rectangle is *n*^{th} rectangle is

Draw five rectangles using right endpoints as shown in Figure (1).

Find the width

Here, the upper limit is *b*, the lower limit is *a*, and the number of rectangles is *n*.

Substitute 10 for *b*, 0 for *a* and 5 for *n* in Equation (2).

Refer to Figure (1),

Take the height of the first rectangle’s

Substitute 5 for *n*, 3.2 for

Therefore, the lower estimate using right endpoints for

Draw five rectangles using left endpoints as shown in Figure (2).

The expression to find the upper estimate of areas of *n* rectangles

Here, the left endpoint height of the first rectangle is *n*^{th} rectangle is

Refer to Figure (2),

Take the height of the left endpoint

Substitute 5 for *n*, 5 for

Therefore, the upper estimate using left endpoints for

(b)

To determine

The lower estimate and upper estimate of the area using ten rectangles.

Expert Solution

The lower estimate of the area using ten rectangles is 14.4.

The upper estimate of the area using ten rectangles is 19.4.

**Given information:**

Take the curve as

Consider that the region lies between

Consider the number of rectangles to be

Use the right end points of the curve as the lower estimate and the left end points of the curve as the upper estimate for the decreasing behavior of the curve (*f*).

Draw ten rectangles using right endpoints as shown in Figure (3).

Substitute 10 for *b*, 0 for *a* and 10 for *n* in Equation (2).

Refer to Figure (3),

Take the height of the right endpoint

Substitute 10 for *n*, 4 for

Therefore, the lower estimate using the right endpoints for

Draw ten rectangles using left endpoints as shown in Figure (4).

The expression to find the upper estimate of the areas of the 10 rectangles

Here, the lower estimate using the right endpoints for

Refer to Figure (4),

Take the height of the left endpoint height

Substitute 14.4 for

Therefore, the upper estimate using the left endpoints for