# How do you find the volume bounded by #y=3-x^2# and y=2 revolved about the y=2?

Volume

(Sorry if this is in excessive detail)

The given regions in the Cartesian plane look like:

If this is rotated about

we have:

For ease of calculation, it is convenient if we shift this solid down so the axis of rotation falls along the X-axis:

Note that the radius (relative to the X-axis) of this shifted volume is

equal to

and we can slice this solid into thin disks, each with a thickness of

so that each disk has a volume of

With very small values of

We can evaluate this sum with

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To find the volume bounded by the curves ( y = 3 - x^2 ) and ( y = 2 ) revolved about the line ( y = 2 ), you can use the method of cylindrical shells.

- Determine the limits of integration by finding the points of intersection between the curves ( y = 3 - x^2 ) and ( y = 2 ).
- Set up the integral for the volume using the formula ( V = \int_{a}^{b} 2\pi x f(x) , dx ), where ( f(x) ) represents the distance between the curves and ( x ) varies from the leftmost intersection point to the rightmost.
- Evaluate the integral to find the volume of the solid generated by revolving the region between the curves about the line ( y = 2 ).

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To find the volume bounded by the curves ( y = 3 - x^2 ) and ( y = 2 ) revolved about the line ( y = 2 ), you can use the method of cylindrical shells.

- Determine the limits of integration by finding the points of intersection between the curves. Set ( 3 - x^2 = 2 ) and solve for ( x ) to find the points of intersection.
- Set up the integral using the formula for the volume of cylindrical shells: ( V = \int_{a}^{b} 2\pi rh , dx ), where ( r ) is the distance from the axis of rotation to the shell, and ( h ) is the height of the shell.
- Express ( r ) and ( h ) in terms of ( x ).
- Integrate with respect to ( x ) from the lower limit of integration ( a ) to the upper limit of integration ( b ).

This will give you the volume of the solid formed by revolving the region bounded by the curves about the line ( y = 2 ).

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