6 The diagram shows the curve with equation \(y = \sqrt { 100 - 4 x ^ { 2 } }\), where \(x \geqslant 0\).
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}
- Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis, giving your answer in terms of \(\pi\).
- Use the mid-ordinate rule with five strips of equal width to find an estimate for \(\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
- The point \(P\) on the curve has coordinates \(( 3,8 )\).
- Find the gradient of the curve \(y = \sqrt { 100 - 4 x ^ { 2 } }\) at the point \(P\).
- Hence show that the equation of the tangent to the curve at the point \(P\) can be written as \(2 y + 3 x = 25\).
- The shaded regions on the diagram below are bounded by the curve, the tangent at \(P\) and the coordinate axes.
\includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731}
Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.