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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 5.1, Problem 2E

(a)

(i)

To determine

The upper estimate of the area under the graph using six rectangles.

Expert Solution

The upper estimate of the area under the graph using six rectangles is 86.6.

**Given information:**

The curve as

The region lies between

Number of rectangles

The sample points are the right endpoints for the lower estimate, the left end points for the upper estimate, and the mid end points for the mid estimate.

**Calculation:**

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

Here, the left 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*.

Substitute 12 for *b*, 0 for *a* and 6 for *n* in Equation (2).

Draw six rectangles using left endpoints as shown in Figure (1).

Refer to Figure (1),

Take the left endpoint height of the first rectangle

Substitute 6 for *n*, 9 for

Therefore, the upper estimate using the left endpoints for

(ii)

To determine

The lower estimate of the area under the graph using six rectangles.

Expert Solution

The lower estimate of the area under the graph using six rectangles is 71.

Draw six rectangles using the right endpoints as shown in Figure (2).

The expression to find the lower estimate of the areas of 6 rectangles

Here, the upper estimate using the left endpoints for

Refer to Figure (2).

Take the right endpoint height of the left uppermost rectangle

Substitute 86.6 for

Therefore, the lower estimate using the left endpoints for

(iii)

To determine

The mid estimate of the area under the graph using six rectangles.

Expert Solution

The mid estimate of the area under the graph using six rectangles is 79.6.

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

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

Draw six rectangles using mid endpoints as shown in Figure (3).

Refer to Figure (3).

Take the mid height of the first rectangle

Substitute 6 for *n*, 9 for

Therefore, the mid estimate using mid endpoints for

(b)

To determine

Whether

Expert Solution

Refer to part (i).

The function

The upper estimate

Hence, the upper estimate

(c)

To determine

Whether

Expert Solution

Refer to part (ii),

The curve is a decreasing curve.

The lower estimate

Hence, the lower estimate

(d)

To determine

The best estimate.

Expert Solution

Refer to part (b) and part (c).

The upper estimate is an overestimate of the true area and the lower estimate is an underestimating of the true area.

Refer to Figure (3).

The mid estimate of the area using mid end points shows the area of each rectangle which appears closer to the true area.

Hence, the mid estimate using mid points seems to be the best estimate.