- (a) Given that \(a\) is a positive constant, use the substitution \(x = a \sin ^ { 2 } \theta\) to show that
$$\int _ { 0 } ^ { a } x ^ { \frac { 1 } { 2 } } \sqrt { a - x } \mathrm {~d} x = \frac { 1 } { 2 } a ^ { 2 } \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$
(b) Hence use algebraic integration to show that
$$\int _ { 0 } ^ { a } x ^ { \frac { 1 } { 2 } } \sqrt { a - x } d x = k \pi a ^ { 2 }$$
where \(k\) is a constant to be found.