Question 1188866
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 *[illustration Vol_by_Cylinder_y_x_3.jpg]

Volume of a cylindrical shell *[tex \Large V\ =\ 2\pi rh\Delta r]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ V_{rot}\ =\ 2\pi\,\int_0^1\,x\(1\,-\,x^3\)\,dx]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ =\ 2\pi\,\int_0^1\,x\,-\,x^4\,dx]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ =\ 2\pi\[\int_0^1\,x\,dx\ -\ \int_0^1\,x^4\,dx\]]


You can do your own arithmetic.

																
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
*[illustration darwinfish.jpg]

From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> 
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