Calculate the area of the shape bounded by the lines y = x ^ 2, x + y-2 = 0.

Let’s equate the equations of the lines and find the points of intersection of their graphs:

x + y – 2 = 0;

y = 2 – x.

x ^ 2 = 2 – x;

x ^ 2 + x – 2 = 0

x12 = (-1 + – √ (1 – 4 * (-2)) / 2 = (-1 + – 3) / 2.

x1 = (-1 – 3) / 2 = -2; x2 = (-1 + 3) / 2 = 1.

Then the required area S will be equal to the difference of the integrals:

S = ∫ (2 – x) * dx | -2; 1 – (∫x ^ 2 * dx | -2; 1 = (2x – x ^ 2/2) | -2; 1 – 1/3 * x ^ 3 | -2; 1 = 8 1/2 – 7/3 = 51/6 – 14/6 = 37/6 = 6 1/6.