A curve has polar equation \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
Sketch the curve.
Find the area of the region enclosed by the curve and the line AB .
Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).