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\textbf{Multiple Integration}

\textit{\textbf{Iteration of Double Integrals}}
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\textbf{Question}

Find the volume of the given solid

Below $z=x^2\sin (y^4)$ and over the triangle in the $xy$-plane
defined by the vertices $(0,0)$, $(0, \pi^{1/4})$ and $(\pi^{1/4},
\pi^{1/4})$.


\textbf{Answer}

\begin{eqnarray*}
V & = & \int_0^{\pi^{1/4}} \,dy \int_0^y x^2 \sin (y^4) \,dx\\
& = & \frac{1}{3} \int_0^{\pi^{1/4}} y^3 \sin (y^4) \,dy\\
\textrm{Let } u & = & y^4\\
du & = & 4y^3dy\\
& & \\
\Rightarrow V & = & \frac{1}{12} \int_0^{\pi} \sin u \,du\\
& = & \frac{1}{6} \textrm{cu. units}
\end{eqnarray*} 

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