find the area between the parabola y^2 = x and the line x + y = 2 Solution First, solve: x^2 = x + 2 ==> x^2 - x - 2 = 0 ==> (x - 2)(x + 1) = 0 ==> x = -1 and x = 2. These intersection points are our limits. We know that x^2 < x + 2 on [-1, 2]. So the integral is: ? x + 2 - x^2 dx (from -1 to 2) = -x^3/3 + x^2/2 + 2x (evaluated from -1 to 2) = [(- 2^3)/3 + (2^2)/2 + 2(2)] - [-(-1)^3/3 + (-1)^2/2 + 2(-1)] = 9/2. <== ANSWER.