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\huge{Week 5 Recitation Problems} \\
\Large{MATH:114, Recitations 309 and 310} \\
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\large{\textbf{Volumes}} \\
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1. Suppose the functions $f(x)$ and $g(x)$ bound a closed region $R$ in the plane. Rotate $R$ around the $x$ axis to get a solid of rotation $S_R$. How does the \textbf{washer} method find the volume of $S_R$? Use words or pictures to explain, including relevant geometric formulas or ideas.
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2. Let $f(x) = x^2$ and $g(x)=x+2$, and let $R$ be the closed region bounded by $f(x)$ and $g(x)$. Find the volume of the solid generated by rotating $R$ around the $x$ axis.
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3. Let the functions $p(x)$ and $q(x)$ bound a closed region $C$ in the plane. Rotate $C$ around the $x$ axis to get a solid of rotation $S_C$. How does the \textbf{shell} method find the volume of $S_C$? Use words or pictures to explain, including relevant geometric formulas.
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4. Why might it be difficult to use the shell method with the functions $f(x)$ and $g(x)$ from Problem 2? \textit{(Hint: how do we find the inverse of $f(x)$?)}
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5. Let $p(x)=x^2$ and $q(x)=-x^4$. Set up an integral to find the volume of the solid found by rotating the region bounded by $p(x)$, $q(x)$, and the vertical line $x=1$ around the $y$ axis. If you have time, compute this integral!
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