Definite Integration Question 211
Question: Area of the ellipse $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $ is
[Karnataka CET 1993]
Options:
A) $ \pi ab $ sq. unit
B) $ \frac{1}{2}\pi ab $ sq. unit
C) $ \frac{1}{4}\pi ab $ sq. unit
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Since the given equation contains only even powers of x and only even powers of y, the curve is symmetrical about y-axis as well as x-axis. Whole area of given ellipse $ =4(\text{area }ofBCO)=4\times \int_0^{a}{ydx=4\int_0^{a}{\frac{b}{a}\sqrt{a^{2}-x^{2}}}dx} $
$ =4ab\int_0^{\pi /2}{( \frac{1+\cos 2\theta }{2} )d\theta } $ , {Putting $ x=a\sin \theta $ } $ =2ab( \int_0^{\pi /2}{d\theta +\int_0^{\pi /2}{\cos 2\theta d\theta }} ) $
$ =[\theta ]_0^{\pi /2}+[ \frac{\sin 2\theta }{2} ]_0^{\pi /2}=\pi ab $ sq. unit.
BETA
