4 The curve with equation \(y = x ^ { 4 } - 8 x + 9\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-5_410_609_383_721}
The point \(( 2,9 )\) lies on the curve.
- Find \(\int _ { 0 } ^ { 2 } \left( x ^ { 4 } - 8 x + 9 \right) \mathrm { d } x\).
- Hence find the area of the shaded region bounded by the curve and the line \(y = 9\).
- The point \(A ( 1,2 )\) lies on the curve with equation \(y = x ^ { 4 } - 8 x + 9\).
- Find the gradient of the curve at the point \(A\).
- Hence find an equation of the tangent to the curve at the point \(A\).