Question

Find the volume of the described solid S: the base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are squares.

Answer 1

Due to symmetry we can integrate from 0 to r and just double the integral
\[V = 2\int\limits_{0}^{r}A(x) dx\]
\[V = 8\int\limits_{0}^{r}(r^{2}-x^{2}) dx\]
\[8 | r^{2}x - x^{3}/3 = 8(r^{3} - r^{3}/3) = \frac{16}{3}r^{3}\]