How do you use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region #y = 16 - x^2# and #0 ≤ x ≤ 4# rotated about the x-axis?

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Answer 1 · 2016-10-28T09:24:39

The volume is 546.1

The volume of a shell is surface area multiplied by thickness
The surface area is #piy^2#
And the volume is =surface area x thickness

#dV=piy^2dx#
#y=16-x^2#
so #y^2=256-32x^2+x^4#

The volume is #V=piint_0^4(256-32x^2+x^4)dx#

#V=pi(256x-(32x^3)/3+x^5/5)_0^4#
#V=pi(256*4-32*64/3+1024/5)#
#V=pi(1024-2048/3+1024/5)=546.1#