1.28: Volume of Revolution about the y-axis

4PM1/2R/June/2019 — Question 9 · 9 marks

The finite region R\displaystyle R enclosed by the y\displaystyle y-axis, the straight line with equation
y+2x=13y+2x=13
and the curve with equation
y=x2−2y=x^2-2
is defined for points with coordinates (x,y)\displaystyle (x,y) with x⩾0\displaystyle x\geqslant0.
The region R\displaystyle R is rotated through 360∘\displaystyle 360^\circ about the y\displaystyle y-axis.
Use algebraic integration to find the volume of the solid generated.
Give your answer in terms of π\displaystyle \pi.
(9)