Chapter 5.3, Problem 27E

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

Use the Midpoint Rule with
*y*-axis the region under the curve

To determine

**To estimate:**

The volume obtained by rotating about the y-axis the region under the given curve.

Explanation

**1) Concept:**

i. If

ii. By the shell method, the volume of the solid by rotating the region under the curve

where,Â

iii. To approximate the integral is the same as finding the Riemann sum for the midpoints. Use the formula and find the value of the integral.

**2) Formula:**

The Riemann sum for the midpoints:

where,

**3) Given:**

**4) Calculation:**

As the region is bounded by

Draw the region using the given curves.

The graph shows the region and the height of cylindrical shell formed at x by the rotation about the line

It has the radius

Therefore, the circumference is

It is given that

So,

So, the volume of given solid is

Assume that

where

For the midpoint rule,

Find

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