Question 1019b

Definite Integrals: Areas and Volumes

Question

A curve has parametric equations

\begin{align*} & x = a ( 2\sin\theta - \cos 2 \theta ), \\ & y = a ( 2\cos\theta - \sin 2 \theta ), \end{align*}

for

{0 \leq \theta \leq 2 \pi.}

(i)

Sketch

{C}

and state the Cartesian equation of its line of symmetry.

[2]

(ii)

Find the values of

{\theta}

at the points where

{C}

meets the

{x\textrm{-axis}.}

[2]

(iii)

Show that the area enclosed by the

{x\textrm{-axis},}

and the part of

{C}

above the

{x\textrm{-axis},}

is given by

\int_{\theta_1}^{\theta_2} a^2 (4 \cos^2 \theta + 2 \cos \theta \sin 2 \theta - 2 \sin^2 2 \theta) \; \mathrm{d} \theta,

where

{\theta_1}

and

{\theta_2}

should be stated

[3]

(iv)

Hence find, in terms of

{a,}

the exact total area enclosed by

{C.}

[5]