find volume of shell y=x y=x^2 over x axis

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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find volume of shell y=x y=x^2 over x axis

Expert Solution
Step 1

This is equivalent to rotating R2 about the line y=0.

Since neither of the curves are touching the axis we are rotating about, we must use washers.

For washers, the volume is given by

V=π∫ba(R(x))2−(r(x))2 dx

Where R(x) is the curve farthest away from our axis of rotation - the top function y=x and r(x) is the curve closest to the axis of rotation - the bottom function y=x2.

As you found the intersecting points, we can set up our integral as

V=π∫0 to 1(x)2−(x2)2 dx

V=π∫0 to  1(x²−x⁴) dx 

=π(x³/3-x⁵/5)0 to 1

=π(1/3 -1/5)

=2π/15

 

 

 

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