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{\bf Exam Question

Topic: Double Integral}

Evaluate the double integral $$\int \!\!\! \int_R
x^2\sin\left(x^4+2x^2y^2+y^4\right)\, d(x,y),$$ where $R$ is the
region satisfying $x^2+y^2\le1$ and $y\ge0.$

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{\bf Solution}

In polar coordinates
\begin{eqnarray*}
I&=&\int_0^{\pi}\, d\theta\int_0^1 r^2\cos^2\theta\sin(r^4)\, dr\\
&=&\int_0^{\pi}\cos^2\theta\, d\theta\int_0^1 r^3\sin(r^4)\, dr\\
&=&\frac{1}{2}\left[\theta+\frac{\sin\theta}{2}\right]_0^{\pi}
\frac{1}{4}\left[-\cos(r^4)\right]_0^1\\
&=&\frac{\pi}{2}.\frac{1}{4}(1-\cos1)
\end{eqnarray*}

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