Question

Find the area bounded by the given curves; y= 1/(1+x^2), x=1,x=-1. sketch the curve indicating the area bounded.

Answer 1

$y= 1/(1+x^2), x=1,x=-1$

Black lines indicating area of bounded region.

Area (A)

$=\int_{-1}^{1} \frac{1}{1+x^2}dx=[\tan^{-1} x]_{-1}^{1}=\tan^{-1} [1]-tan^{-1} [-1]\\ =\frac{\pi}{4}-(-\frac{\pi}{4})=\frac{\pi}{4}+\frac{\pi}{4}=\frac{\pi}{2}$