   Chapter 7.2, Problem 63E

Chapter
Section
Textbook Problem

# Find the volume obtained by rotating the region bounded by the curves about the given axis. y = sin x ,   y = cos x ,   0 ≤ x ≤ π / 4 ; 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=sinx,y=cosx

Horizontal limit 0xπ4

Axis of rotation: y=1

Calculation:

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

rin=1cosx,

rout=1sinx

Substitute the curves into the volume formula with a and bas 0 and π4 respectively;

V=0π4π(r2outr2in)dx=0π4π((1sinx)

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