1.19: Past-paper question 9

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y^2 = x - 1$ and part of the line $\displaystyle l$ with equation $\displaystyle 2y + x - 4 = 0$

The region $\displaystyle R$, bounded by the $\displaystyle x$-axis, the curve $\displaystyle C$ and the line $\displaystyle l$, is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

Using algebraic integration, find the exact value of the volume of the solid generated.