The area bounded by the curves y = ln x ,  y = ln x ,  y = ln  x and y = ln x , for x ∈ ( − 1 ,   1 ) is

Step-by-step solution

The required shaded region is as shown in figure As the graph is symmetric in all quadrants, we calculate area in one quadrant and multiply by 4. Hence, required area = 4 | ∫ 0 1 ( l n x ) . d x | A = 4 | [ x l n x − x ] 0 1 | A = 4 | [ ( 1 l n 1 − 1 ) − lim x → 0 + ( x l n x − x ) ] | A = 4 | [ ( 0 − 1 ) − lim x → 0 + l n x 1 x − 0 ] | = 4 | [ − 1 − lim x → 0 + 1 x − 1 x 2 ] | A = 4 | [ − 1 + lim x → 0 + x ] | A = + 4 | − 1 | A = 4