The area of the region between the curves y= 1+ x x and y= 1- x x bounded by the lines x=0 and x= 4 is

Step-by-step solution

aligned & Required area = ₀^ / 4 ( 1+ x x - 1- x x ) d x [ 1+ x x > 1- x x >0 ] & .= ₀^ / 4 1+ 2 x 2 1+ ^2 x 2 1- ^2 x 2 1+ ^2 x 2 1- 2 x 2 1+ ^2 x 2 1- ^2 x 2 1+ ^2 x 2 ) d x & .= ₀^ / 4 1+ x 2 1- x 2 - 1- x 2 1+ x 2 ) d x & aligned = ₀^ / 4 1+ x 2 -1+ x 2 1- ^2 x 2 d x= ₀^ / 4 2 x 2 1- ^2 x 2 d x Put x 2 =t aligned 1 2 ^2 x 2 d x & =d t Required area & = ₀^ 8 4 t d t (1+t^2 ) 1-t^2 & = ₀^ 2 -1 4 t (1+t^2 ) 1-t^2 d t [ astan 8 = 2 -1 ] aligned