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Math Help - Not sure of the procedure to find the volume of a solid for a function

  1. #1
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    Not sure of the procedure to find the volume of a solid for a function

    Hi all,

    I'm looking at a bonus for an assignment:

    Compute the volume of the solid produced when the region between the x-axis and \[f(x) = \frac{1}{x}\], for \[1 \le x \le \infty \], is revolved about the x-axis.

    I've never done these types of problems, but I set it up like this:

    \[\begin{array}{l}<br />
\int_{1\,}^{\,\infty } {{x^{ - 1}}} {\rm{d}}x\\<br />
\mathop {\lim }\limits_{t \to \infty } \int_{1\,}^{\,t} {{x^{ - 1}}} {\rm{d}}x\\<br />
\left. {\mathop {\lim }\limits_{t \to \infty } \left[ {\ln x} \right]} \right|_{1\,}^{\,t}\\<br />
\mathop {\lim }\limits_{t \to \infty } \left[ {\ln t - \ln 1} \right]\\<br />
\mathop {\lim }\limits_{t \to \infty } \left[ {\ln t - 0} \right]\\<br />
\ln \infty \\<br />
\infty <br />
\end{array}\]

    Kind of as I expected in my head...the horizontal asymptote on the x-axis would mean that this function is divergent with that infinite limit. Based on that, I wouldn't be able to calculate any volume.

    I was wondering if my thinking was right on this or if there is a different procedure I need to take? I simply haven't done any volume problems before. Am I out in left field or is my thinking right?

    Thank you in advance!
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  2. #2
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    What you're being asked is to evaluate a solid of revolution.

    You need to imagine the region under the graph as a series of rectangles. These rectangles get rotated to form cylinders.

    The cylinders have a radius the same as the value of the graph at that point, so the radius of each cylinder is \displaystyle y = \frac{1}{x}. Therefore the area of each circular cross-section is \displaystyle \pi \left(\frac{1}{x}\right)^2 = \frac{\pi}{x^2}.

    We will call the height of each cylinder \displaystyle dx, which is a small change in \displaystyle x.

    So the volume each cylinder is \displaystyle \frac{\pi}{x^2}\,dx and the volume of the solid can be approximated by \displaystyle \sum{\frac{\pi}{x^2}\,dx}.

    As you increase the number of cylinders and make the height of each cylinder \displaystyle \to 0, this sum converges on a definite integral, and the approximation becomes exact.

    So \displaystyle V = \int_1^{\infty}{\frac{\pi}{x^2}\,dx}.

    You should find that this integral converges...
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  3. #3
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    That makes a lot more sense, I get it. I knew there had to be something wrong with my thinking. That looks a lot better. I can take it from there, and go study that subject a bit. Your steps to reach that integral makes sense to me, though.

    Thank you so much!!!!
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