# 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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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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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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