Chapter 5.1, Problem 5E

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8th Edition

James Stewart

ISBN: 9781305270343

Textbook Problem

(a) Estimate the area under the graph of *f*(*x*) = 1 + *x*^{2} from *x* = −1 to *x* = 2 using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.

(b) Repeat part (a) using left endpoints.

(c) Repeat part (a) using midpoints.

(d) From your sketches in parts (a)–(c), which appears to be the best estimate?

**(a)**

To determine

**To find:**

The area under the graph using right endpoints and three rectangles and then area under the graph using right endpoints and six rectangles.

Explanation

**Given:**

The curve function is

The region lies between

**Calculation:**

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 three rectangles with right end points.

Substitute 2 for *b*, -1 for *a,* and 3 for *n* in Equation (2).

From Figure (1), take the right endpoint height of the first rectangle’s

Substitute 3 for *n*, 1 for

(b)

To determine

The area under the graph using left endpoints and three rectangles and then area under the graph using left endpoints and six rectangles.

(c)

To determine

The area under the graph using midpoints and three rectangles and then area under the graph using midpoints and six rectangles.

(d)

To determine

**To find:**

The best estimate.

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