Question

P(2, 3) lies on the curve y=(1/2)x^2+1. O is the origin, M is the foot of the perpendicular to the x-axis and the curve intersects the y-axis at A. The area bounded by AO, OM, MP and the curve PA of the curve is rotated through four right angles about the y-axis. Find the volume of the solid formed, giving your answer as a multiple of π.

Answer 1

$$y=\dfrac{x^2}{2}+1$$

$$V=2\pi\displaystyle\int_{0}^{2}x(\dfrac{x^2}{2}+1)dx=2\pi\bigg[\dfrac{x^4}{8}+\dfrac{x^2}{2}\bigg]\begin{matrix} 2 \\ 0 \end{matrix}$$

$$=8\pi(units^3)$$

$$Answer:8\pi$$