Calculate the area of the bounded line shape y = -x ^ 2 + 4 and y = x ^ 2-2x.

Find the intersection points of the graphs, for this we equate the equations of the functions to each other:

-x ^ 2 + 4 = x ^ 2 – 2x;

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 area of the figure formed by the given lines will be equal to the difference of the integrals:

S = ∫ (-x ^ 2 + 4) * dx | -1; 2 -∫ (x ^ 2 – 2x) * dx | -1; 2 = (-1 / 3x ^ 3 + 4x) | -1; 2 – (1 / 2x ^ 2 – x) | -1; 2 = (-8/3 + 8 – 1/3 – 4) – (2 – 2 – 1/2 + 1) = 1 – 1/2 = 1 / 2.

Answer: the required area S is 1/2.