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\begin{center}
  18.022 Recitation Quiz \iftoggle{solutions}{(with solutions)}{} \\
  26 November 2014 \\
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\itm (6.2.12 in \textit{Colley}) Use Green's theorem to find the area enclosed by the hypocycloid $\vec{x}(t) =(a\cos^3t, a\sin^3t),$ $0\leq t \leq 2\pi$. (We found the length of the hypocycloid on 15 October). Feel free to leave an unevaluated definite integral in your answer. 

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  \includegraphics{figures/hypocycloid}
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\sol{
Green's theorem implies that the area of a region $D$ is given by $\oint_{\partial D} x\,dy$. We calculate  
\begin{align*}
  \oint_{\partial D} x\,dy &= \int_0^{2\pi} (a\cos^3t)(3a\sin^2t\cos t)\,dt \\
  &= 3a^2 \int_0^{2\pi} \cos^4 t \sin^2 t\,dt = \boxed{3\pi a^2/8}. \qedhere
\end{align*}
}{\vfill} 

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