Example 1 | Volumes of Solids of Revolution

$V = \pi {\displaystyle {\int_{x_1}}^{x_2}} {y_U}^2 \, dx$

$V = \pi {\displaystyle {\int_0}^2} (x^{1/2})^2 \, dx$

$V = \pi {\displaystyle {\int_0}^2} x \, dx$

$V = \pi \left[ \dfrac{x^2}{2} \right]_0^2$

$V = \frac{1}{2}\pi [ \, 2^2 - 0^2 \, ]$

$V = 2\pi \, \text{ unit}^3$           answer

$V = 2\pi {\displaystyle {\int_{y_1}}^{y_2}} y_C(x_R - x_L) \, dy$

$V = 2\pi {\displaystyle {\int_0}^\sqrt{2}} y(2 - y^2) \, dy$

$V = 2\pi {\displaystyle {\int_0}^\sqrt{2}} (2y - y^3) \, dy$

$V = 2\pi \left[ \dfrac{2y^2}{2} - \dfrac{y^4}{4} \right]_0^\sqrt{2}$

$V = 2\pi \left[ (\sqrt{2})^2 - \dfrac{(\sqrt{2})^4}{4} \right]$

$V = 2\pi \, \text{ unit}^3$           answer