Two crossing pillars

Idea. Slice parallel to both axes. Each cylinder cuts the slicing plane in a band, and the two bands cross at right angles, so every slice is a square. Integrating the square's area gives the volume.

Step 1: the slices. Put the axes along the $x$ and $y$ directions, crossing at the origin. The first cylinder is $y^2+z^2\le 1$ and the second $x^2+z^2\le 1$. At height $z$, the first gives $|y|\le \sqrt{1-z^2}$ and the second $|x|\le \sqrt{1-z^2}$: a square of side $2\sqrt{1-z^2}$.

Step 2: integrate. $$V={\int }_{-1}^{1}4(1-z^2)dz=4\left(2-\frac{2}{3}\right)=\frac{16}{3}.$$

A comparison with the ball. The unit ball, which fits inside the intersection, has slices that are discs inscribed in those squares, of area $\frac{\pi }{4}$ times the square's. So the ball's volume is $\frac{\pi }{4}\cdot \frac{16}{3}=\frac{4\pi }{3}$, as it should be. Archimedes knew this solid; it is called the Steinmetz solid.