Fig. 1 illustrates a section of the curve $y = x^2 - 1$ together with the line $y = h$, where $h$ is a constant.
(a(i))[3]
The shaded region is turned through $360^{\circ}$ around the $y$-axis. Show that the volume of revolution, $V$, is $V = \pi\left(\frac{1}{2}h^2 + h\right)$.
(a(ii))[4]
Find the area of the shaded region when $h = 3$, showing all working needed.
(b)[4]
Find the rate at which the height of the water level is increasing when the height of the water level is $3$ cm.
Worked solution & mark scheme
This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Required volume integral $V=\pi\int (y+1)\,dy$” …