14 The curve \(C\) is defined for \(t \geq 0\) by the parametric equations
$$x = t ^ { 2 } + t \quad \text { and } \quad y = 4 t ^ { 2 } - t ^ { 3 }$$
\(C\) is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-26_691_608_541_717}
14
- Find the gradient of \(C\) at the point where it intersects the positive \(x\)-axis.
14 - The area \(A\) enclosed between \(C\) and the \(x\)-axis is given by
$$A = \int _ { 0 } ^ { b } y \mathrm {~d} x$$
Find the value of \(b\).
14
- (ii) Use the substitution \(y = 4 t ^ { 2 } - t ^ { 3 }\) to show that
$$A = \int _ { 0 } ^ { 4 } \left( 4 t ^ { 2 } + 7 t ^ { 3 } - 2 t ^ { 4 } \right) \mathrm { d } t$$
14
- (iii) Find the value of \(A\).