The area enclosed by the curve y = -x^2 and the line x + y + 2 = 0 is

Step-by-step solution

Find the points of intersection of the curve y = -x^2 and the line x + y + 2 = 0 . Substituting y = -x^2 into the line equation: x - x^2 + 2 = 0 x^2 - x - 2 = 0 (x - 2)(x + 1) = 0 The points of intersection are at x = -1 and x = 2 . In the interval [-1, 2] , the parabola y = -x^2 lies above the line y = -x - 2 . The required area is given by: A = _ -1 ² [(-x^2) - (-x - 2)] dx A = _ -1 ² (-x^2 + x + 2) dx A = [ - x^3 3 + x^2 2 + 2x ]_ -1 ² A = ( - 8 3 + 4 2 + 4 ) - ( 1 3 + 1 2 - 2 ) A = ( 6 - 8 3 ) - ( 5 6 - 2 ) A =