Example 2 | Volumes of Solids of Revolution

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

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

$V = \pi {\displaystyle {\int_0}^\sqrt{2}} [ \, 4 - y^4 \, ] \, dy$

$V = \pi \left[ 4y - \dfrac{y^5}{5} \right]_0^\sqrt{2}$

$V = \pi \left[ 4(\sqrt{2}) - \dfrac{(\sqrt{2})^5}{5} \right]$

$V = 14.22 \, \text{ unit}^3$           answer

$V = 2\pi {\displaystyle {\int_{x_1}}^{x_2}} x_C(y_U - y_L) \, dx$

$V = 2\pi {\displaystyle {\int_0}^2} x(x^{1/2} - 0) \, dx$

$V = 2\pi {\displaystyle {\int_0}^2} x^{3/2} \, dx$

$V = 2\pi \left[ \dfrac{x^{5/2}}{5/2} \right]_0^2$

$V = \frac{4}{5}\pi \, [ \, 2^{5/2} - 0^{5/2} \, ]$

$V = 14.22 \, \text{ unit}^3$           answer