Chapter 5.3, Problem 8E

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8th Edition

James Stewart

ISBN: 9781285740621

Textbook Problem

Let *V* be the volume of the solid obtained by rotating about the *y*-axis the region bounded by
*V* both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.

To determine

**To find:**

The volume of the solid obtained by rotating the region bounded by the given curves about

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. If the cross section is the disc and the radius of the disc is in terms of

iv. The volume of the solid revolution about the

**2) Given:**

The region bounded by

**3) Calculation:**

As region is bounded by

i. By the slice method,

From the figure, as the region rotates about

Find the intersection of the two curves by solving the simultaneous equation,

Therefore, the curves

A cross section has the shape of a washer with the inner radius

So, find the area of cross-sectional by subtracting the area of the inner circle and the outer circle:

Therefore, the volume of the solid of revolution about

By using the fundamental theorem of calculus and the power rule of integration,

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