PX275 - B4d - volumes of revolution

revolving the plane around the $x$ -axis to obtain a solid, cylindrical volume
taking the area element, $d A$ , and revolving it by an angle, $d θ$ , around the $x$ -axis
the volume element:

d V = d A y d θ = y d θ d x d y

V = \int_{x = x_{1}}^{x_{2}} \int_{θ = 0}^{2 π} d θ \int_{y = 0}^{f (x)} y d y d x = 2 π \int_{x_{1}}^{x_{2}} \frac{(f (x))^{2}}{2} d x

∴ V = π \int_{x_{1}}^{x_{2}} (f (x))^{2} d x

example

\begin{aligned} V & = π \int_{0}^{h} {(\frac{a x}{h})}^{2} d x \\ = π {(\frac{a}{h})}^{2} {[\frac{x^{3}}{3}]}_{0}^{h} \\ ∴ V & = \frac{1}{3} π a^{2} h \end{aligned}