A curve $C$ has parametric equations $x=a\left( \frac{4}{\pi }t+\sin 2t \right)+2a$, $y=a\cos t$, wh

Question

A curve $C$ has parametric equations $x=a\left( \frac{4}{\pi }t+\sin 2t ight)+2a$, $y=a\cos t$, where $-\frac{\pi }{2}\le t\le \frac{\pi }{2}$ and $a$ is a positive constant. Sketch $C$. State clearly the coordinates of any points where $C$ meets the $x$-axis. [2] Show that the area enclosed by $C$ and the $x$-axis is given by ${{a}^{2}}\displaystyle\int_{{{	heta }_{1}}}^{{{	heta }_{2}}}{\left( \frac{4}{\pi }\cos t+2\cos t\cos 2t ight)\mathrm{d}t}$ where ${{	heta }_{1}}$ and ${{	heta }_{2}}$ are values to be stated. [3] Hence find, in terms of $a$, the exact area enclosed by $C$ and the $x$-axis. [3]

Accepted Answer

(i) $(0,0)$ and $(4a,0)$ (ii) $\theta_1=-\dfrac\pi2$, $\theta_2=\dfrac\pi2$ (iii) $a^2\left(\dfrac8\pi+\dfrac43\right)\text{ units}^2$

2021 HCI P2 Q1

2021 HCI P2 Q1

Video Solution:

Solution:

Solution locked