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\textbf{Multiple Integration}

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

Sketch the domain of integration, and calculate the iterated integral
for
$$\int_0^1 \,dy \int_y^1 e^{-x^2} \,dx$$


\textbf{Answer}

$$ \ $$
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\begin{eqnarray*}
I &amp; = &amp; \int_0^1 \,dy \int_y^1 e^{-x^2} \,dx\\
&amp; = &amp; \int_R e^{-x^2} \,dx\\
&amp; = &amp; \int_0^1 e^{-x^2} \int_0^x \,dy\\
&amp; = &amp; \int_0^1 xe^{-x^2} \,dx\\
\textrm{Let } u &amp; = &amp; x^2\\
\Rightarrow du &amp; = &amp; 2xdx\\
&amp; &amp; \\
\Rightarrow I &amp; = &amp; \frac{1}{2} \int_0^1 e^{-u}\\
&amp; = &amp; \left. -\frac{1}{2} e^{-u} \right |_0^1\\
&amp; = &amp; \frac{1}{2} \left ( 1 - \frac{1}{e} \right )
\end{eqnarray*}


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