1.25: Past-paper question 10

Figure 3 shows part of the curve $\displaystyle C$ with equation $\displaystyle y = 3\sqrt{4-x}$ and part of the line $\displaystyle l$ with equation $\displaystyle px + qy + r = 0$ where $\displaystyle p$ and $\displaystyle r$ are integers and $\displaystyle q$ is prime.

The curve $\displaystyle C$ cuts the $\displaystyle y$-axis at the point $\displaystyle A$

The line $\displaystyle l$ is the tangent to $\displaystyle C$ at $\displaystyle A$

(a) Find the value of $\displaystyle p$, the value of $\displaystyle q$ and the value of $\displaystyle r$

The finite region $\displaystyle R$, shown shaded in Figure 3, is rotated through $\displaystyle 360^\circ$ about the $\displaystyle x$-axis.

(b) Use algebraic integration to find the volume of the solid formed.
Give your answer in terms of $\displaystyle \pi$