   Chapter 7.2, Problem 64E

Chapter
Section
Textbook Problem

# Find the volume obtained by rotating the region bounded by the curves about the given axis. y = sec x ,   y = cos x ,   0 ≤ x ≤ π / 3 ; about y = − 1

To determine

To evaluate: volume of the solid generated by rotating the region bounded by the given curves about the given axis.

Explanation

Consider a two function f(x) and g(x) between the points x=a and x=b. If the region between the two functions between a and b is rotated about x-axis, a solid shape is obtained. The volume of that solid shape can be calculated by taking the volume of small circular“washers”making up the solid and then adding them together.

Formula used:

Volume of the solid obtained by rotating the region between the curves f(x) and g(x) from x=a to b where f(x)g(x) is given by the following integral:

V=abπ((f(x))2(g(x))2)dx,           where 0a<b

Given:

The curves,y=secx,y=cosx

Horizontal limit 0xπ3

Axis of rotation: y=1

Calculation:

The axis of rotation is y=1. The inner and outer radius of the washer will be:

rin=1+cosx,

rout=1+secx

Substitute the curves into the volume formula with a and bas 0 and π3 respectively;

V=0π3π(r2outr2in)dx=0π3π((1+secx)2(1+cosx)2</

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