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

How do you find the volume of #y=x^2# going from [0, 2] on the x-axis and [0, 4] on the y-axis?

Answer 1

Ezekiel Alexander

#6pi# cubic units

Volume = #pi int x^2dy#, with y from 2 to 4 and x from #y= x^2#.

#=pi int ydy,# with y from 2 to 4

#=pi[y^2/2],# between the limits 2 and 4

#=pi/2(4^2-2^2)=6pi# cubic units.

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Answer 2

Emma Aldridge

To find the volume of the solid formed by revolving the region bounded by ( y = x^2 ) on the interval ([0, 2]) on the x-axis and ([0, 4]) on the y-axis about the x-axis, you can use the method of cylindrical shells.

The volume ( V ) is given by:

[ V = 2\pi \int_a^b x \cdot f(x) , dx ]

Where ( f(x) = x^2 ), ( a = 0 ), and ( b = 2 ).

Substitute the values into the formula and integrate to find the volume.

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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.