# How do you find the volume of a solid that is enclosed by #y=x^2#, #y=0#, and #x=2# revolved about the x axis?

The volume is

A unit slice has a volume

Therefore,

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To find the volume of the solid enclosed by (y = x^2), (y = 0), and (x = 2) revolved about the x-axis, we can use the method of cylindrical shells.

The formula to find the volume using cylindrical shells is:

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

where ( f(x) ) is the function that forms the outer boundary of the solid (in this case, ( y = x^2 )), and ( a ) and ( b ) are the x-values where the solid begins and ends, respectively (in this case, ( a = 0 ) and ( b = 2 )).

So, we have:

[ V = 2\pi \int_{0}^{2} x \cdot x^2 , dx ]

[ V = 2\pi \int_{0}^{2} x^3 , dx ]

[ V = 2\pi \left[ \frac{x^4}{4} \right]_{0}^{2} ]

[ V = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) ]

[ V = 2\pi \left( \frac{16}{4} \right) ]

[ V = 2\pi \cdot 4 ]

[ V = 8\pi ]

So, the volume of the solid enclosed by (y = x^2), (y = 0), and (x = 2) revolved about the x-axis is (8\pi) cubic units.

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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.

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.

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.

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