Volumes of Revolution Cross Sections

ChemE14
Wed, 28 Apr 2010 23:05:40 +0000

This is a very quick explanation (though it’s been discussed in more detail in other AB/BC threads on here).

Rotational Volume (I’ve talked more about this on at least one other BC thread, so you can check my recent posts to find those links):
Disk method
V = pi * integral from a to b [R(x)]^2 dx, where R is the distance from the function to the axis of revolution

Washer method
V = pi * integral from a to b [ [R(x)]^2 - [r(x)]^2 ] dx, where R is the bigger distance from the axis of revolution and r is the smaller distance from the axis of revolution

Volume by Cross-Sections:
V = [integral from a to b] A(x)dx, where A is the cross-sectional area. It’s pretty easy once you get the hang of it. The problem will tell you the shape of the cross-sections (semicircle, square, rectangle with a certain height, triangle, equilateral triangle, etc.) and, if applicable, the height of those cross-sections (the height can be constant or in terms of x). The base of the cross-sections is the distance from the upper-y-bound (this is usually the function) to the lower-y-bound (this is usually a horizontal line).

Example: f(x) = x^2, where region R is bounded by x=3, x=1, and y=0. Find volume with semicircle cross-sections coming out perpendicular to the x-axis.

1) Identify your equation for cross-sectional area
What is the area of a semicircle? Half the area of circle, so
A = pi * r^2 / 2
So now we must identify ‘r’, or the radius of the semicircle. You see that the base of each semicircle is the distance from f(x) to y=0, right? So the base = f(x) - 0 = f(x). However, this base represents the diameter of the semicircle - for the equation, we need the radius, so

r = d/2 = f(x)/2

Now plug this back into the equation to get

A(x) = pi * [f(x)/2]^2 / 2 = (pi/2) * [f(x)]^2 / 4 = (pi/8) * [f(x)]^2

All you have to do now is integrate this from x=1 to x=3, and you have your answer.

Your book may have a list of specific equations for each cross-sectional shape, and while you can memorize them, I personally think it’s just easier to derive the cross-sectional volume equations yourself.