Find the area of the figure y = x-x ^ 2 at y = 0. find the zeros of the function.

Find the intersection points of the graph of the function with the oX axis, for this we equate the equation to zero:

x – x ^ 2 = 0;

x * (1 – x) = 0;

x1 = 0; x = 1.

Then the area of the figure S bounded by the given lines will be equal to the integral:

S = ∫ (x – x ^ 2) * dx | 0; 1 = (x ^ 2/2 – x ^ 3/3) | 0; 1 = (1 ^ 2/2 – 1 ^ 3/3) – (0 ^ 2/2 – 0 ^ 3/3) = (1/2 – 1/3) – 0 = 1/6.

Answer: the required area is 1/6.