6 A curve has equation \(y = x ^ { 5 } - 2 x ^ { 2 } + 9\). The point \(P\) with coordinates \(( - 1,6 )\) lies on the curve.
- Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
- The point \(Q\) with coordinates \(( 2 , k )\) lies on the curve.
- Find the value of \(k\).
- Verify that \(Q\) also lies on the tangent to the curve at the point \(P\).
- The curve and the tangent to the curve at \(P\) are sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{aa42b4fd-1e37-48b8-90ee-269916c4db2c-4_721_887_936_589}
- Find \(\int _ { - 1 } ^ { 2 } \left( x ^ { 5 } - 2 x ^ { 2 } + 9 \right) \mathrm { d } x\).
- Hence find the area of the shaded region bounded by the curve and the tangent to the curve at \(P\).
(3 marks)