How do you find the volume of the solid obtained by rotating the region bounded by #y=x# and #y=x^2# about the #x#-axis?

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Answer 1 · 2018-04-24T12:27:27

#V=(2pi)/15#

First we need the points where #x# and #x^2# meet.

#x=x^2#

#x^x-x=0#

#x(x-1)=0#

#x=0 or 1#

So our bounds are #0# and #1#.

When we have two function for the volume, we use:
#V=piint_a^b(f(x)^2-g(x)^2)dx#

#V=piint_0^1(x^2-x^4)dx#

#V=pi[x^3/3-x^5/5]_0^1#

#V=pi(1/3-1/5)=(2pi)/15#