How do you deal with absolute values in a function when calculating volumes of solids of revolution? For example, when calculating the volume of a solid of revolution for the area between the curves `y=-abs(x-4)+4` ; `y=0` about the x axis. I know the equation should be set up as `pi * integral between 0 and 8 of (-abs(x-4)+4)^2 dx` , but am unsure of how to proceed thereafter.

I've done this volume calculation using Pappus's $(2^{nd})$ Centroid Theorem: the volume of a planar area of revolution is the product of the area $A$ and the length of the path traced by its centroid R, i.e., $2πR$. The volume is simply $V=2\pi RA$.

The figure below shows the area to be revolved about the $x$-axis. The centroid of a triangle is straightforward and is shown here by the asterisk $x_c,y_c=4,4/3$ and the area is simply $A=16$.

Hence,

$$V=2\pi\frac{4}{3}16=\frac{128\pi}{3}$$

I've verified this result numerically by an alternative method. Even if this isn't the method you want, you'll have a (verified) result to compare.

![enter image description here](