Let the area of the region A = (x,y) : 2^x y 2 + x - x^2, x 0 be p q - ₂(e) , where p and q are coprime natural numbers. Then the value of p + q is equal to

Step-by-step solution

The bounding curves are y = 2^x and y = 2 + x - x^2 . To find their intersection in the domain x 0 , we equate them: 2^x = 2 + x - x^2 By observation, x = 1 satisfies the equation since 2^1 = 2 + 1 - 1^2 = 2 . In the interval (0, 1) , the parabola y = 2 + x - x^2 lies above the exponential curve y = 2^x . The required area A is given by the definite integral: A = ₀¹ (2 + x - x^2 - 2^x) dx Integrating the terms: A = [ 2x + x^2 2 - x^3 3 - 2^x 2 ]₀¹ Evaluating at the upper bound x = 1 : ( 2 + 1 2 - 1 3 - 2 2 ) = 13 6