$${\begin {array} {l} V = \int\limits_{-r}^r 2 \pi (r^2 - x^2) dx = \pi \cdot \left (r^2x - \frac {1} {3} x^3 \right) \Bigg |_{-r}^r \\= \pi \left (r^3 - \frac {1} {3}r^3 \right) - \pi \left ( - r^3 + \frac {1} {3}r^3 \right) = \frac {4} {3} \pi r^3 \end {array}}$$