MHB Bob's question at Yahoo Answers regarding mimimizing a solid of revolution

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The discussion focuses on finding the minimum volume of the solid of revolution formed by rotating the function f(x) = p/x^p around the x-axis over the interval [1, infinity). Using the disk method, the volume is expressed as V = πp²/(2p-1) for p > 1/2. Differentiating V with respect to p reveals a critical point at p = 1, where the volume is minimized. The analysis shows that for 1/2 < p < 1, the volume decreases, while for p > 1, it increases, confirming that the minimum occurs at p = 1. Therefore, the minimum volume of the solid is achieved when p equals 1.
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Here is the question:

Solid of Revolution Calculus 2 question.?

For a positive real number p, define f(x)=p/x^(p). Find the minimum value of the volume of the solid created by rotating this function around the x-axis over the interval [1,infinity).

I have posted a link there to this thread so the OP can view my work.
 
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Hello bob,

First, let's write the given function as:

$$f(x)=px^{-p}$$

Now, using the disk method, we find that the volume of an arbitrary disk is:

$$dV=\pi r^2\,dx$$

where:

$$r=f(x)=px^{-p}$$

And so we have:

$$dV=\pi p^2 x^{-2p}\,dx$$

Summing us all of the disks, we may state:

$$V=\pi p^2\int_1^{\infty}x^{-2p}\,dx$$

This is an improper integral with the unbounded upper limit, so we may write:

$$V=\pi p^2\lim_{t\to\infty}\left(\int_1^{\infty}x^{-2p}\,dx \right)$$

Applying the FTOC, we have:

$$V=\pi p^2\lim_{t\to\infty}\left(\left[\frac{x^{1-2p}}{1-2p} \right]_1^{t} \right)$$

$$V=\frac{\pi p^2}{1-2p}\lim_{t\to\infty}\left(\frac{1}{t^{2p-1}}-1 \right)$$

For $$\frac{1}{2}<p$$ we have:

$$V(p)=\frac{\pi p^2}{2p-1}$$

To determine the critical value(s), we need to differentiate with respect to $p$ and equate the result to zero. Using the quotient rule, we find:

$$V'(p)=\frac{(2p-1)(2\pi p)-\left(\pi p^2 \right)(2)}{(2p-1)^2}=\frac{2\pi p(p-1)}{(2p-1)^2}$$

For $$\frac{1}{2}<p$$ we have:

$$p=1$$

If we observe that for $$\frac{1}{2}<p<1$$ we have $V'(p)<0$ and for $1<p$ we have $V'(p)>0$, we may therefore conclude that this critical value is at a minimum.

Thus, we may conclude that the described solid of revolution is minimized when $p=1$.
 
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