\begin{array}{l} V = \int_{- r}^{r} 2 π (r^{2} - x^{2}) d x = π \cdot (r^{2} x - \frac{1}{3} x^{3}) |_{- r}^{r} \\ = π (r^{3} - \frac{1}{3} r^{3}) - π (- r^{3} + \frac{1}{3} r^{3}) = \frac{4}{3} π r^{3} \end{array}

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

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