question_answer

If \[{{c}_{1}}=y=\frac{1}{1+{{x}^{2}}}\] and \[{{c}_{2}}=y=\frac{{{x}^{2}}}{2}\] be two curves lying in XY-plane, then

A) Area bounded by curve \[y=\frac{1}{1+{{x}^{2}}}\] and \[y=0\] is \[\frac{\pi }{2}\]

B) Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[\frac{\pi }{2}-1\]

C) Area bounded by \[{{c}_{1}}\] and \[{{c}_{2}}\] is \[1-\frac{\pi }{2}\]

D) Area bounded by curve \[y=\frac{1}{1+{{x}^{2}}}\] and x-axis is \[\frac{\pi }{2}\]

Correct Answer: B

Solution :

[b] Area bounded by \[y=\frac{1}{1+{{x}^{2}}}\] and x - axis is \[\int_{-\infty }^{\infty }{\frac{1}{1+{{x}^{2}}}dx=\pi }\] Area bounded by two curves is \[\int_{-1}^{1}{\left( \frac{1}{1+{{x}^{2}}}-\frac{{{x}^{2}}}{2} \right)dx=\frac{\pi }{2}-1}\]