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\includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-4_563_679_938_733}
The diagram shows points \(A ( 0,4 )\) and \(B ( 2,1 )\) on the curve \(y = \frac { 8 } { 3 x + 2 }\). The tangent to the curve at \(B\) crosses the \(x\)-axis at \(C\). The point \(D\) has coordinates \(( 2,0 )\).
- Find the equation of the tangent to the curve at \(B\) and hence show that the area of triangle \(B D C\) is \(\frac { 4 } { 3 }\).
- Show that the volume of the solid formed when the shaded region \(O D B A\) is rotated completely about the \(x\)-axis is \(8 \pi\).