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\begin{center}
\textbf{Multiple Integration}

\textit{\textbf{Double Integrals}}
\end{center}

\textbf{Question}

Evaluate the following double integral by inspection.

$\int \! \int_{|x|+|y|\le1} \left ( x^3 \cos (y^2) +3\sin y -\pi
\right ) \,dA$


\textbf{Answer}

$\begin{array}{l}
\int \! \int_{|x|+|y|\le1} \left ( x^3 \cos (y^2) +3\sin y -\pi
\right ) \,dA\\
= 0 + 0 -\pi(\textrm{area bounded by }|x|+|y|=1) \\
= -\pi \times 4 \times \frac{1}{2}(1)(1) =2\pi
\end{array}
 \ \ \
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\end{array}$

(Each of the first two terms in the integrand is an odd function of
one of the variables, and also the square is symmetrical about each of
the coordinate axes.)


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