- In this question you must show detailed reasoning.
Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 2 } { x ^ { 2 } - x + 1 } \mathrm {~d} x\). Give your answer in exact form.
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\section*{8. In this question you must show detailed reasoning.}
The diagram shows the curve with equation \(y = \frac { x + 3 } { \sqrt { x ^ { 2 } + 9 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{f42517a5-d7ed-40f3-bb04-faea97d4b19b-18_890_1010_367_242}
The region R , shown shaded in the diagram, is bounded by the curve, the \(x\)-axis, the \(y\)-axis, and the line \(x = 4\).
- Determine the area of R . Give your answer in the form \(p + \ln q\) where \(p\) and \(q\) are integers to be determined.
The region R is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Determine the volume of the solid of revolution formed. Give your answer in the form \(\pi \left( a + b \ln \left( \frac { c } { d } \right) \right)\) where \(a , b , c\) and \(d\) are integers to be determined.
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