1.11: Past-paper question 4

The curve $\displaystyle S$ with equation $\displaystyle y = \frac{x^2}{4} + 2$ where $\displaystyle x \geq 0$ and the line $\displaystyle l$ with equation $\displaystyle 2y - x - 4 = 0$ where $\displaystyle x \geq 0$ intersect at the points $\displaystyle A$ and $\displaystyle B$, as shown in Figure 2.

(a) (i) Show that the coordinates of point $\displaystyle A$ are $\displaystyle (0, 2)$

(ii) Find the coordinates of the point $\displaystyle B$

The finite region bounded by $\displaystyle S$ and $\displaystyle l$, shown shaded in Figure 2, is rotated through $\displaystyle 2\pi$ radians about the $\displaystyle y$-axis.

(b) Use algebraic integration to find the volume of the solid generated.

Give your answer in terms of $\displaystyle \pi$