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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

y = tan x ,   y = 0 ,   x = π / 4

(a) About the x-axis

(b) About y = 1

To determine

Part (a):

i. To set up:

An integral for the volume of solid obtained by rotating the region bounded by the given curves about the x axis.

ii. $2:

The integral by using the calculator up to five decimal places.

Explanation

1) Concept:

i. If the cross section is a disc and the radius of the disc is in terms of x  or y then area A=π radius2

ii. The volume of a solid revolution about the  x-axis is

V= abA(x)dx

2) Given:

y=tanx, y=0, x=π4, about the x-axis

3) Calculation:

Bounded region of the given curves is as shown below:

From the above graph

The solid lies between x=0 and  x=π4

 The region rotates about the x-axis, so the cross-section is perpendicular to the

To determine

Part (b):

i. To set up:

An integral for the volume of solid obtained by rotating the region bounded by the given curves about the line y= -1.

ii. To evaluate:

The integral by using the calculator up to five decimal places

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