   Chapter 7.3, Problem 38E

Chapter
Section
Textbook Problem

# Find the volume of the solid obtained by rotating about the line x = 1 the region under the curve y = x 1 − x 2 ,   0 ≤ x ≤ 1.

To determine

To Find: The Volume of the resulting solid

Explanation

Calculation: We use Cylindrical Shell method to find the Volume

The surface area of one cylinder is A=2πrh

V=ab2πrhdx

The height h is given by the curve x1x2 and the radius is r=1x because we’re rotating about x=1

V=2π01(1x)(x1x2)dx=2π[01x1x2dx01(x21x2)dx]

Consider x1x2dx

Let 1x2=t2,2xdx=2tdtxdx=tdt

x1x2dx=t.tdt=t2dt=t33=(1x2)323

Therefore

V=2π[[13(1x2)32]0101x21x2dx]

=2π[[13(01)]01x21x2dx]

=2π[1301x21x2dx]

We Know that

sinnudu=1nsinn1ucosu+n1nsinn2udu

sin2x=12(1cos2x)

Consider solving the second integral

x=sinudx=cosudu1x2=1sin2u=cosu

01x21x2dx

=sin2u(cosu)(cosudu)=sin2u

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