Question 1 - A-Level Maths

A curve has polar equation \(r = a \cos 3 \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), where \(a\) is a positive constant.
1. Sketch the curve, using a continuous line for sections where \(r > 0\) and a broken line for sections where \(r < 0\).
2. Find the area enclosed by one of the loops.
Find the exact value of \(\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x\).
Use a trigonometric substitution to find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + 3 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).