Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=x^2 , x=y^2; about y=1 Why is the area: A(x)=Pi[(1-x^2)^2-(1-sqrt(x))^2]? Whis is...

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

y=x^2 , x=y^2; about y=1

Why is the area: A(x)=Pi[(1-x^2)^2-(1-sqrt(x))^2]?

Whis is there 1 - X^2 coming from?

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mlehuzzah eNotes educator | Certified Educator

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First the volume of a cylinder is

`pi R^2 h`

If we were to drill a hole through the middle of the cylinder, such that we now had a tube-like thing, or a washer, the volume would be the volume of the original cylinder minus the volume of the hole, or:

`pi R^2 h - pi r^2 h = pi (R^2-r^2) h`

 

 

Ok, I don't know how well you can see the various lines in the picture, but here goes:

The black tear drop shape gets rotated around the dotted black line.  If you were to slice this shape with vertical slices, you would get a bunch of washers: cylinders with cylindrical holes in them.  The formula for the volume of this shape is:

`int_0^1 pi (R^2 - r^2)...

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