Definite Integration Question 177
Question: Area bounded by the curve $ xy^{2}=a^{2}(a-x) $ and y-axis is
Options:
A) $ \pi a^{2}/2 $ sq. units
B) $ \pi a^{2} $ sq. units
C) $ 3\pi a^{2} $ sq. units
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
[b] $ xy^{2}=a^{2}(a-x)\Rightarrow x=\frac{a^{3}}{y^{2}+a^{2}} $ The given curve is symmetrical about x-axis, and meets it at (a, 0).
The line $ x=0 $ , i.e., y-axis is an asymptote. Area $ =\int\limits_0^{\infty }{xdy}=2\int\limits_0^{\infty }{\frac{a^{3}}{y^{2}+a^{2}}dx} $
$ =2a^{2}\frac{1}{a}[ {{\tan }^{-1}}\frac{y}{a} ]_0^{\infty } $
$ =2a^{2}\frac{\pi }{2}=\pi a^{2} $ sq. units
BETA
