- A curve, which is part of an ellipse, has parametric equations
$$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } .$$
The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Show that the area of the surface generated is given by the integral
$$k \pi \int _ { 0 } ^ { \alpha } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \quad \text { where } c = \cos \theta$$
and where \(k\) and \(\alpha\) are constants to be found.
- Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.