\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}

Calculate the given double integrals by iteration

$\ds \int\!\!\!\int_T \frac{xy}{1+x^4} \,dA$

With $T$ being the triangle with vertices $(0,0)$, $(1,0)$ and
$(1,1)$.


\textbf{Answer}

\begin{eqnarray*}
I & = & \int\!\!\!\int_T \frac{xy}{1+x^4} \,dA\\
 & = & \int_0^1 \frac{x}{1+x^4} \,dx \int_0^x y \,dy\\
& = & \frac{1}{2} \int_0^1 \frac{x^3}{1+x^4} \,dx\\
& = & \frac{1}{8} \left. \ln (1+x^4) \right |_0^1 = \frac{\ln 2}{8}
\end{eqnarray*}

\end{document}



