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Find the volume $V$ of the solid obtained by rotating the region bounded by the given curves about the specified line.

$y = \ln{7x}$;

$y = 3$

$y = 4$

$x = 0$

Rotated about the $Y$-Axis.

How would I approach solving this problem. I'm stuck. Thanks for the help!

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Since we need to rotate about the $Y$-axis, we are forming infinitesimal disks which are aligned on the $y$-axis. (Imagine that the $y$-axis is a rod that goes through each center of a thin disk). These disks go from $y=3$ to $y=4$. The $x=0$ is just the $y$-axis.

Geometrically, these thin disks have a radius perpendicular to the $y$-axis and so to find this radius, we can solve $y=\ln{7x}$ as a function of $x$. By doing this, we can use the distance from the $y$-axis to the curve as the radius. (Go ahead and turn your paper counter-clockwise to show the "heights" from the $y$-axis to the function. These "heights" are the radii of your disks.

We solve for $x$ and get: $$y=\ln{7x}$$ $$e^y=e^{\ln{7x}}$$ $$e^y=7x$$ $$x=\frac{1}{7}e^y$$

We are summing up the volume of all these REALLY thin disks. So we are adding each infinitesimal disk between $3$ and $4$ on the $y$-axis. This is the same as integrating a volume function from $3$ to $4$ of an area function at each point. Since rotating this gives us a circle, $\pi{r^2}$.

Our function $x=\frac17e^y$ is now a function that outputs a radius. (Quite convenient but not a coincidence!)

So we integrate and get $$V = \int_3^4{\pi(\frac17e^y)^2}dy$$ $$V = \frac{\pi}{49}\int_3^4e^{2y}dy$$ $$V= \frac{\pi}{49}\left[\frac12e^{2y}\right]_3^4$$ $$V= \frac{\pi}{98}\left(e^8-e^6\right)\approx82.628$$

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Let's first review what is a solid of revolution and the two types of methods of finding volumes of solids of revolution. To generate a solid of revolution, we need to have a generating region and a axis of revolution. In your case, the generating region is the region bounded by a curve and three straight lines and the axis of revolution is the y-aixs.

As a non-math undergrad, the notes written by Prof. Dawkins helped me a lot out of this: http://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx

So we have the method of rings and method of cylinders. Method of rings cuts the generating region into thin rectangles which are perpendicular to the revolving axis; method of cylinders cuts the generating region into thin rectangles which are parallel to the revolving axis. Their differences are illustrated below: Difference between the two methods

This difference is extremely important in deciding which one to use. In the above case, you will opt for method of cylinder. It will be easier to find out the length of the each rectangle. However, in some cases, both methods would do.


Here is the cookbook for your problem:

  1. Plot $y=ln(7x)$, $y=3$, $y=4$, $x=0$ on the same plane to sketch for the generating region.

  2. Decide how are you cutting the region. Horiziontally? or Vertically? I recommend to cut horizontally in this case (i.e. the length of rectangle is parallel to the x axis). Why?

  3. Choose the appropriate method for finding the volume (method of disks or method of cylinders) according to your choice in 2. (Which one is it in this case?)

  4. Determine the integrand by considering the respective formulas for each method. In your case, you may have to express x in terms of y in order to find the height each rectangle.

  5. Perform integration.

Hope it helps! Feel free to post a follow up if you are stuck in any or the steps! I believe Ian has already give a detailed procedure on steps 4 and 5.

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