How do you find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=x^2, y=1, about y=2?

1 Answer
Dec 21, 2017

See below.

Explanation:

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First we find the volume of the area A+B and then subtract the volume of the area B to give us the volume of the area A:

Volume A+B:

It can be see from the diagram that the radius cd of volume A+B is #2-x^2# so the integral will be:

#V=pi*int_(-1)^(1)(2-x^2)^2 dx#

#(2-x^2)^2=4-4x^2+x^4#

#V=pi*int_(-1)^(1)(4-4x^2+x^4) dx=[4x-4/3x^3+1/5x^5]_(-1)^(1)#

#[4x-4/3x^3+1/5x^5]^(1)-[4x-4/3x^3+1/5x^5]_(-1)#

Plugging in upper and lower bounds:

#V=pi*[4(1)-4/3(1)^3+1/5(1)^5]^(1)-[4(-1)-4/3(-1)^3+1/5(-1)^5]_(-1)#

#V=pi*[4-4/3+1/5]^(1)-[-4+4/3-1/5]_(-1)#

#V=pi*[43/15]^(1)-[-43/15]_(-1)=86/15pi#

Volume of B:

This produces a cylinder of radius ab= 1 and length ( this is the length of the interval #[ -1 , 1 ]# which is 2:

#V=pi(1)^2(2)=2pi#

Volume of A= volume(A+B)- volumeB =#(86pi)/15-2pi=color(blue)((56pi)/15)# units cubed.

Volume of A:

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