Question

The area enclosed by the curve $x=aco{s}^{3}t,,y=bsi{n}^{3}t$ and the positive directions of x-axis and y-axis is

• A. $\frac{\pi ab}{4}$
• B. $\frac{\pi ab}{32}$
• C. $\frac{3\pi ab}{32}$
• D. $\frac{5\pi ab}{32}$

Answer 1

Answer: (C)

$y=0$, when $t=0$ and then $x=a$
So desire area
$A=\underset{0}{\overset{a}{\int }}ydx=\underset{\pi /2}{\overset{0}{\int }}bsi{n}^{3}t\left(-3aco{s}^{2}tsintdt\right)$
$A=3ab\underset{0}{\overset{\pi /2}{\int }}si{n}^{4}tco{s}^{2}tdt=3ab\underset{0}{\overset{\pi /2}{\int }}co{s}^{4}tsi{n}^{4}tdt$
$\therefore 2A=3ab\underset{0}{\overset{\frac{\pi }{2}}{\int }}co{s}^{2}tsi{n}^{2}tdt=3ab\cdot \frac{\pi }{16}\Rightarrow A=\frac{3\pi ab}{32}$