Volume of intersections of cylinders

See also Steinmetz Solid .

If one were to set this up as an integral, you'd realize how similar it is to the volume of a sphere. The only difference is the area of the largest cross-section (or the equatorial area.

If you looked at the volume of a sphere of radius 1, with equatorial cross-sectional area of $\pi$ , is $\frac{4}{3}\pi$ . The volume is just $\frac{4}{3}$ times the equatorial cross-sectional area.

By analogy, you can also get the volume of the intersection region as the area of the "equatorial" cross-section, which is 4, times $\frac{4}{3}$ , which gives $\frac{16}{3}$ .