Home > Calculus > Determining the Volume of a Solid of Revolution

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?

Answer 1

William Abdalla

#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#

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Answer 2

Samantha Adkison

To find the volume of the solid obtained by rotating the region bounded by ( y=x ) and ( y=x^2 ) about the x-axis, you can use the method of cylindrical shells.

The volume ( V ) is given by the integral:

[ V = \int_{a}^{b} 2\pi x (x-x^2) , dx ]

where ( a ) and ( b ) are the x-coordinates of the intersection points of ( y=x ) and ( y=x^2 ).

Solve for ( a ) and ( b ) by setting ( x = x^2 ), then integrate from ( a ) to ( b ). This yields the volume of the solid of revolution.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.