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Find the volume of solid generated when $y=2 \sin(x^2)$ is rotated about the $x$ axis. The function intersects the $x$-axis at 0 and $\sqrt\pi$.

$$ \text{ Volume } = \int_0^\sqrt\pi \pi(2 \sin x^2)^2 dx=\int_0^\sqrt\pi 4\pi( \sin x^2)^2dx.$$

However I cannot figure out how to evaluate this integral. I tried substitution method and integration by parts with no luck. Any help is much appreciated.

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    $\begingroup$ Are you sure you're supposed to integrate this by hand? $\endgroup$ – Andrew Li Feb 21 '18 at 2:02
  • $\begingroup$ Yes. This was a question on the study guide and we are not allowed to use any online integral calculator. $\endgroup$ – nova_star Feb 21 '18 at 2:10
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    $\begingroup$ The solution is $-\pi^{3/2}(-2 + C[2]) =8.4179$, where $C[z] = \int_{0}^{z} \cos(\pi t^{2}/2) dt$ is the FresnelC integral. So I doubt, how one can do this by hand. $\endgroup$ – Nash J. Feb 21 '18 at 2:13
  • $\begingroup$ I got the same answer using an online integral calculator. I was hoping there is an easier way to do this. But perhaps not. Thanks for your help. $\endgroup$ – nova_star Feb 21 '18 at 2:18
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$4π\int sin²(x²) dx$ $2π\int 1dx - 2π\int 1-2sin²(x²)dx$ $2πx-2π\int cos(2x²)dx$

$u=\sqrt{2}x$ $du=\sqrt{2}dx$

$2πx-\sqrt{2}π\int cos(u²)du$ $2πx-\sqrt{2}πC(u)+c$ where C is the Cosine Fresnel Integral $2πx-\sqrt{2}πC(\sqrt{2}x)+c$

$[2πx-\sqrt{2}πC(\sqrt{2}x)]_0^\sqrt{π}$

$2π\sqrt{π}-\sqrt{2}πC(\sqrt{2π})$

if it’d work better, there’s also a method you can use where instead of using fresnel integrals, you just add up error functions with complex arguments. Comment below if you’d like me to show you.

Hope this helps 😊

EDIT: the person who asked the question requested that I use the aforementioned error dunction method. Let’s go back to this step:

$2πx-\sqrt{2}π\int cos(u²)du$

$cos(ø)=\frac{e^{øi}+e^{-øi}}2$

$2πx-\sqrt{2}π\int \frac{e^{u²i}+e^{-u²i}}2 du$ $2πx-\frac{π}{\sqrt{2}}\int e^{u²i}du-\frac{π}{\sqrt{2}}\int e^{-u²i}du$ $2πx-\sum \frac{π}{\sqrt{2}}\int e^{±u²i}du$

$v=\frac{(1∓i)u}{\sqrt{2}}$ $u=\frac{(1±i)v}{\sqrt{2}}$ $du=\frac{1±i}{\sqrt{2}}dv$

$2πx-\sum \frac{π}{\sqrt{2}}\int e^{±(\frac{(1±i)v}{\sqrt{2}})²i}•\frac{1±i}{\sqrt{2}}dv$ $2πx-\sum \frac{π(1±i)}{2}\int e^{-v²}dv$

$\int e^{-v²}dv=\frac{2erf(v)}{\sqrt{π}}$

$2πx-\sum \frac{π(1±i)}{2}•\frac{2erf(v)}{\sqrt{π}}$ $2πx-\sum \sqrt{π}(1±i)erf(\frac{(1∓i)u}{\sqrt{2}})$ $2πx-\sqrt{π}(1+i)erf(\frac{(1-i)u}{\sqrt{2}})-\sqrt{π}(1-i)erf(\frac{(1+i)u}{\sqrt{2}})$

$u=\sqrt{2}x$

$2πx-\sqrt{π}(1+i)erf((1-i)x)-\sqrt{π}(1-i)erf((1+i)x)+C$

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  • $\begingroup$ I am totally unfamiliar with fresnel integral. Just read up on it. I will really appreciate it if you can show me how to add up error functions with complex arguments. $\endgroup$ – nova_star Feb 22 '18 at 12:15

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