Question

The curve \(y = ax - x^2\) cuts the \(x\)-axis at the points \((0,0)\) and \((a,0)\). The area enclosed between the curve and the \(x\)-axis is \(36\) square units. Calculate the value of \(a\).

With this value of \(a\) find, as a multiple of \(\pi\), the volume obtained when the above area is rotated through \(4\) right angles about the \(x\)-axis.