   Chapter 5.R, Problem 12E

Chapter
Section
Textbook Problem

# Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = tan x ,   y = x ,   x = π / 3 ; about the y -axis

To determine

To set up:

An integral for the volume of a solid.

Explanation

1) Concept:

If x  is the radius of a typical shell, then circumference=2πx and height is y=f(x).

By shell method, the volume of the solid obtained by rotating the region under the curve y=f(x) from a to b about y- axis is

V= ab2πxf(x)dx;axb

where 2πx  is circumference, fx is height, and dx  is the thickness of the shell.

2) Given:

y=tanx,y=x,x= π3;

3) Calculation:

The equation of given curves is

y=tanx, y=x

The region and typical shell are shown in the figure.

The graph shows the region and height of cylindrical shell formed by rotation about the y-axis.

Using shell method, find the typical approximating shell with radius  x

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