# Thread: Not sure of the procedure to find the volume of a solid for a function

1. ## 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:

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

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?

2. 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...

3. 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!!!!