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\begin{center}{ \Large{Worksheets}  \\\large{10/12/2017} }\end{center}
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\begin{enumerate}
\item Let $R$ be the triangle with vertices (1,0), (1,1) and (3,1). Fill in the blanks
\[ \iint_R f(x,y)dA =  \intop_{}^{} ~~ \intop_{}^{} ~~~~~ dydx =  \intop_{}^{} ~~ \intop_{}^{} ~~~~~ dxdy\]
Then compute the double integral over $R$ of the function $f(x,y)=x$.
\vspace{3in}
\item Let $R$ be the region on the $xy$ plane bounded by the graph of $y=x^2$, the $y$-axis, the line $y=1$, in the first quadrant. Fill in the blanks
\[ \iint_R f(x,y)dA =  \intop_{}^{} ~~ \intop_{}^{} ~~~~~ dydx =  \intop_{}^{} ~~ \intop_{}^{} ~~~~~ dxdy\]
Then compute the double integral over $R$ of the function $f(x,y)=x^3\sqrt{1+y^3}$.
\newpage
\item Let $W$ be the solid given by $0\le x\le y\le z\le 1$. Fill in the blanks
\[ \iiint_W f(x,y,z)dV =  \intop_{}^{} \intop_{}^{} \intop_{}^{} ~~~~~~~ dzdydx =  \intop_{}^{} \intop_{}^{}  \intop_{}^{} ~~~~~~~ dxdzdy=  \intop_{}^{} \intop_{}^{}  \intop_{}^{} ~~~~~~~ dydxdz\]
Then compute the triple integral over $W$ of the function $f(x,y,z)=y$.\vspace{3.8in}
\item Sketch the region $W$ bounded by the planes $x=0$, $y=0$, $z=x+y$ and $z=1$. Then compute \[ \iiint_W (x^2+yz)dV\]\newpage
\item Sketch the region $W$ bounded by the planes $x=0$, $y=0$, $z=1$, the paraboloid $z=x^2+y^2$, in the octant $x,y,z\ge 0$. Then compute volume of $W$.
\vspace{4in}
\item Sketch the region $W$ bounded by the planes $x=0$, $y=0$, $z=0$, $z=1$, the cylinder $x^2+y^2=1$, in the octant $x,y,z\ge 0$. Then compute \[ \iiint_W zdV\]
\newpage
\item Compute the length of the cycloid $c(t)=(t-\sin t,1-\cos t)$, $t\in[0,2\pi]$.\vspace{3in}
\item Compute the length of the asteroid $c(t)=(\sin^3 t,\cos^3 t)$, $t\in[0,2\pi]$.\vspace{3in}
\item Compute the length of the graph of $y=x^\frac{2}{3}$ from $x=0$ to $x=8$.
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\end{enumerate}
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