6 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } t ^ { n } \sqrt { 1 + t ^ { 2 } } \mathrm {~d} t\) for integers \(n \geqslant 0\).
- Show that \(I _ { 1 } = \frac { 7 } { 3 }\).
- Prove that, for \(n \geqslant 2 , ( n + 2 ) I _ { n } = 8 ( \sqrt { 3 } ) ^ { n - 1 } - ( n - 1 ) I _ { n - 2 }\).
The curve \(C\) is defined parametrically by
$$x = 10 t ^ { 3 } , y = 15 t ^ { 2 } \text { for } 0 \leqslant t \leqslant \sqrt { 3 }$$
When the curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
- Determine
- the values of the integers \(k\) and \(m\) such that \(A = k \pi I _ { m }\),
- the exact value of \(A\).