The area of the region bounded by the line y = x + 2 and the curve x = -y^2 is

Step-by-step solution

The given curves are x = -y^2 and y = x + 2 . To find the points of intersection, substitute x = y - 2 into the equation of the parabola: y - 2 = -y^2 y^2 + y - 2 = 0 (y + 2)(y - 1) = 0 The points of intersection are at y = -2 and y = 1 . The area of the bounded region is given by integrating the difference of the right curve x = -y^2 and the left curve x = y - 2 with respect to y from y = -2 to y = 1 : A = _ -2 ¹ (-y^2 - (y - 2)) , dy A = _ -2 ¹ (-y^2 - y + 2) , dy Integrating the expression: A = [ - y^3 3 - y^2 2