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I am hoping for some help with this question:

The curve $y=sinx$ where $0\leq x\leq \pi$ is revolved about the line $y=c$ where $0\leq c\leq 1$ to generate a solid

(a)Find a value of $c$ that minimizes the volume of the solid. What is the minimum volume? (b) What value of $c$ in $[0,1]$ maximizes the volume of the solid?

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Hint: Revolving $f(x)$ about the line $y=c$ is the same as revolving $f(x)-c$ about the line $y=0$. – Chris Taylor Feb 2 '12 at 8:32
up vote 0 down vote accepted

The distance from $y=\sin x$ to $y=c$ is the radius of the solid at a particular $x$-coordinate. Hence

$$V=\int_0^\pi \pi(\sin x-c)^2dx.$$

Expand this out and integrate (ignore the constant factor if you want); it's quadratic in $c$ so you should be able to minimize it with ease. (Remember, $dV/dc=0$ is a prerequisite for this.)

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