What rotational volume formula to use

Question

I am taking some parts of calculus 2 and I need to know when one should use the correct rotational formula for volume when rotated around the $x$-axis.

I know that in my course, we only use three formulas for volume, that is:

Rotational volume for circle.

Rotational volume for washers.

Rotational volume for cylindrical shells.

However, how do I know which one to use given a function to rotate around the $x$-axis for? Sometimes we get a function that we should use circle formula for, others we get a function for cylindrical shells. This is not mentioned in the question on the test, so how do I know what to use?

Say I get the function $f(x) = e^x$, and it should be rotated around the $x$-axis for some given interval, how do I know that this is a circle, washer or cylindrical?

Very big thanks if someone could explain.

Answer 1

You use the disc method when you are rotating a function of $x$ around the $x$-axis. The volume of revolution in the interval $[a, b]$ is given by $$V = \pi \int_{a}^{b} [f(x)]^2~dx$$ You are slicing the volume into discs of radius $f(x)$ that are a distance $|x|$ from the $x$-axis.

The disc method can also be used for finding the volume of revolution of a function of $y$ that is rotated around the $y$-axis.

The disc method would be appropriate for your example of rotating $f(x) = e^x$ around the $x$-axis since you are rotating the area between the curve and the $x$-axis around the $x$-axis.

In this case, if you want to find the volume of revolution about the $x$-axis in the interval $[0, 2]$ of the function $f(x) = e^x$, you rotate the region shown in yellow about the $x$-axis. The height of the disc at point $x$ is given by $f(x) = e^x$. The volume is given by $$V = \pi \int_{0}^{2} e^{2x}~dx$$

You use the shell method when you are rotating a function of $x$ around the $y$-axis. The volume of revolution in the interval $[a, b]$ is given by $$V = 2\pi \int_{a}^{b} xf(x)~dx$$ where you are slicing the volume into cylindrical shells of radius $x$ and height $f(x)$.

The shell method can also be used for finding the volume of revolution of a function of $y$ that is rotated around the $x$-axis.

In this case, if you want to find the volume obtained by showing the region shaded in yellow about the $y$-axis, you would evaluate the integral $$V = 2\pi \int_{0}^{2} xe^x~dx$$ where $x$ is the radius and $e^x$ is the height of the cylindrical shell at $x$.

The washer method is used when you wish to find the volume of a hollow solid of revolution. It is a modification of the disc method in which the volume of the inner solid of revolution is subtracted from the volume of the outer solid of revolution. If a region with outer radius $f(x)$ and inner radius $g(x)$ is rotated around the $x$-axis over the interval $[a, b]$, then the volume of revolution is given by $$V = \pi \int_{a}^{b} ([f(x)]^2 - [g(x)]^2)~dx$$ Notice that the disc method is just the special case of the washer method in which the inner radius is $0$.

The washer method can also be used to find a volume of revolution for a function of $y$ that is rotated around the $y$-axis.

For example, if we wish to find the volume of revolution obtained by rotating the area between the curves $f(x) = 3x$ and $g(x) = x^2$ around the $x$-axis, we would evaluate the integral $$V = \pi \int_{0}^{3} [9x^2 - x^4]~dx$$ since the region has outer radius $f(x) = 3x$ and inner radius $g(x) = x^2$.

Answer 2

It is not the case that one is correct and the others are wrong. Assuming you want the volume of the solid of rotation resulting from rotating the graph of a function $f$ between $x=a$ and $x=b$ around the $x$ axis, any of the methods may be applied and, if done correctly, gives the right answer in the end.

The difference between the methods is how exactly they divide your volume into many tiny little pieces that are easy to find the (approximate) volume of. Do you prefer to cut the volume into thin slices that are shaped like coins (flat, solid cylinders), or do you prefer to cut the volume into a series of pipes around the $x$-axis (cylindrical shells)? (The washer method is the former method, only applied to the region between two positive functions; the coins then get a hole in the middle, and become washers).

The practical difference lies in how difficult the resulting integral is to compute (and some times how difficult the correct integrand and integration bounds are to find). And that is something that will differ greatly from function to function. The only general advice I can give is to just try one, see if it works, and if not try another. Practice, and you will get better.