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Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the $y$-axis.

$$y = e^{-x^2},\ y = 0,\ x = 0,\ x = 1$$

How the fudge am I supposed to do this without parts? Is there supposed to be some kind of antiderivative to this function with an $x$ in front of it or something? This is like three functions in one.

Obviously I can get as far as $2\pi\displaystyle\int_0^1xe^{-x^2}dx$ but yeah...

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It looks as if you may have reached the right integral, which is $$\int_0^1 2\pi xe^{-x^2}\, dx.$$ You can quickly integrate by making the substitution $u=x^2$.

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  • $\begingroup$ i've been taking too long of a break in homework... thanks $\endgroup$
    – J L
    Sep 16, 2013 at 2:57
  • $\begingroup$ You are welcome. The term $e^{-x^2}$ is admittedly scary. There is no elementary function whose derivative is $e^{-x^2}$. But the $2x$ in front was very helpful. $\endgroup$
    – André Nicolas
    Sep 16, 2013 at 3:06

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