# The region under the curves #y=x^2, y=x# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?

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a) To sketch the region and find the volume of the solid obtained by rotating the region under the curves ( y = x^2 ) and ( y = x ) about the x-axis:

- Sketch the curves ( y = x^2 ) and ( y = x ) on the same coordinate system within the desired domain.
- Determine the points of intersection by solving ( x^2 = x ) for ( x ).
- Set up the integral for the volume using the formula ( V = \pi \int_{a}^{b} (f(x))^2 - (g(x))^2 , dx ), where ( f(x) ) is the upper curve and ( g(x) ) is the lower curve.
- Integrate the expression over the interval of intersection to find the volume.

b) To find the volume of the solid obtained by rotating the region under the curves ( y = x^2 ) and ( y = x ) about the y-axis:

- Sketch the curves ( y = x^2 ) and ( y = x ) on the same coordinate system within the desired domain.
- Determine the points of intersection by solving ( x^2 = x ) for ( x ).
- Set up the integral for the volume using the formula ( V = \pi \int_{c}^{d} (f(y))^2 - (g(y))^2 , dy ), where ( f(y) ) is the right curve and ( g(y) ) is the left curve.
- Integrate the expression over the interval of intersection to find the volume.

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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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