The area of the region R = (x,y) : 0 y ( x, (e^2 - x)) is

Step-by-step solution

The region is bounded below by y=0 and above by the curves y = x and y = (e^2 - x) . First, find the point of intersection of the two curves: x = (e^2 - x) x = e^2 - x 2x = e^2 x = e^2 2 The corresponding y -coordinate is y = ( e^2 2 ) = 2 - 2 . To find the area, it is easier to integrate with respect to y . We express the curves as functions of y : Left curve: y = x x = e^y Right curve: y = (e^2 - x) e^y = e^2 - x x = e^2 - e^y The required area A is given by: A = ₀^ 2- 2 (x_ right - x_ left ) dy A = ₀^ 2- 2 ((e^2