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I have an $(x,y)$ dataset read in from a file. The data do not follow any sort of equation, but have the general shape of $y=x^2$. The data below are a placeholder for my actual data

data = Table[{x,x^2},{x,0,10,0.1}]

I interpolated the dataset:

f = Interpolation[data, InterpolationOrder->2]

I now have an interpolated function f[x] that can be integrated.

How do I perform an integration to find the volume of this interpolated function rotated about the $y$ axis? For example, if my function were actually $y=x^2$, I would convert $x=\sqrt y$ and set the integral up as $\pi\int_a^by\,\mathrm{d}y$ to perform a disk integral about the $y$ axis. How can I do something equivalent using the interpolated function f?

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1 Answer 1

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You can use InverseFunction.

data = Table[{x, x^2}, {x, 0, 10, 0.1}];
f = Interpolation[data, InterpolationOrder -> 2]

g = InverseFunction[f];
π NIntegrate[g[y]^2, {y, 0, data[[-1, 2]]}]
(* 15708. *)
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