If the area of the region bounded by the curves y = x , y = (a - x) (where a > 2 ), and the x -axis is 2 square units, then the value of a is

Step-by-step solution

The region is bounded by y = x , y = (a - x) , and y = 0 . First, find the point of intersection of the curves: x = (a - x) x = a - x 2x = a x = a 2 The corresponding y -coordinate is y = ( a 2 ) . Expressing the curves as functions of y : Left curve: y = x x = e^y Right curve: y = (a - x) e^y = a - x x = a - e^y The area A is given by integrating with respect to y : A = ₀^ (a/2) (x_ right - x_ left ) dy A = ₀^ (a/2) ((a - e^y) - e^y) dy = ₀^ (a/2) (a - 2e^y) dy Evaluating the integral: A = [ ay - 2e^y ]₀^ (a/2) A