1.17: Past-paper question 6

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = \frac{1}{4x}$, $\displaystyle x > 0$ and part of the curve $\displaystyle S$ with equation $\displaystyle y = 2x^2$, $\displaystyle x \geq 0$

The curve $\displaystyle C$ and the curve $\displaystyle S$ intersect at the point $\displaystyle A$

(a) Find the coordinates of point $\displaystyle A$

The finite region $\displaystyle R$, shown shaded in Figure 3, bounded by the curve $\displaystyle C$, the curve $\displaystyle S$ and the straight line $\displaystyle y = 4$ is rotated through $\displaystyle 360^\circ$ about the $\displaystyle y$-axis.

(b) Find, using algebraic integration, the exact volume of the solid formed.