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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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)...
(The entire section contains 367 words.)
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