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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 5.1, Problem 6E

(a)

To determine

**To draw:**The graph for the function

Expert Solution

The graph for the function

**Given information:**

The curve function is

The region lies in the interval

Draw the graph for the function

(b)

(i)

To determine

The area under the graph of *f* using right endpoints and four rectangles.

Expert Solution

The area under the graph of *f* using right endpoints and four rectangles is 4.44.

Draw the graph for the function

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

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

Find the width

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

Find the area estimate for four rectangles with right end points:

Substitute 5 for *b*, 1 for *a,* and 4 for *n* in Equation (2).

Refer to Figure 2.

Take the right endpoint height of the first rectangle’s

Substitute 4 for *n*, 0.6 for

Therefore, the area under the graph of *f* using right endpoints for

(ii)

To determine

The area under the graph of *f* using midpoints and four rectangles.

Expert Solution

The area under the graph of *f* using midpoints and four rectangles is 3.93.

Draw the graph for the function

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

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

Find the area estimate for four rectangles with mid points:

Refer to Figure 3.

Take the midpoint height of the first rectangle’s

Substitute 4 for *n*, 0.78 for

Therefore, the area under the graph of *f* using mid points for

(c)

(i)

To determine

The area under the graph of *f* using right endpoints and eight rectangles.

Expert Solution

The area under the graph of *f* using right endpoints and eight rectangles is 3.94.

Draw the graph for the function

Find the area estimate for eight rectangles with right end points:

Substitute 5 for *b*, 1 for *a,* and 8 for *n* in Equation (2).

Refer to Figure 4.

Take the right endpoint height of the first rectangle’s

Substitute 8 for *n*, 0.76 for

Therefore, the area under the graph of *f* using right endpoints for

(ii)

To determine

The area under the graph of *f* using midpoints and eight rectangles.

Expert Solution

The area under the graph of *f* using midpoints and eight rectangles is 3.975.

Draw the graph for the function

Find the area estimate for eight rectangles with mid points:

Refer to Figure 5.

Take the midpoint height of the first rectangle’s

Substitute 8 for *n*, 0.87 for

Therefore, the area under the graph of *f* using mid points for