Question

how do i get the volume of the right circular
cone generated by rotating the line y =2/3 x
about the y-axis between y = 0 and y = 3.

Answer 1

well it's and inverted cone



so the volume is

\[V = \pi \int\limits_{0}^{3} x^2 dy\]

so take the equation

y = 2/3x

and make x the subject

so

x = 3/2y

now square it so you get

\[x^2 = \frac{9y^2}{4}\]

now substitute and the integral is

\[V = \pi \int\limits_{y = 0}^{y = 3} \frac{9y^2}{4} dy\]

so just integrate with respect to y.

hope it helps