Solutions - 0030-01

(1) Set up integral.

$$\begin{array}{cc}V& ={\int }_a^{b}2\pi Rhdr\\ & \\ & \gg \gg {\int }_0^{2}2\pi x\left(\frac{1}{\sqrt{x^2+1}}\right)dx\end{array}$$

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(2) Perform $u$-substitution with $u=x^2+1$ and $du=2x$:

$$\begin{array}{c}\gg \gg V={\int }_0^5\frac{\pi }{\sqrt{u}}du\end{array}$$

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(3) Integrate with power rule:

$$\begin{array}{c}\gg \gg \pi {\frac{u^{1/2}}{1/2}|}_0^5\\ \\ \gg \gg 2\pi (\sqrt{5}-1)\end{array}$$