\documentclass[a4paper,12pt]{article}
\usepackage{epsfig}
\newcommand{\un}{\underline}
\newcommand{\ds}{\displaystyle}
\begin{document}
\parindent=0pt

\begin{center}
\textbf{Multiple Integration}

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

\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}

$$ \ $$
\begin{center}
\epsfig{file=MI-2A-1.eps, width=50mm}
\end{center}

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


\end{document}