Calculate the area of a curved trapezoid bounded by lines y = 2x-x2; y = 0.

Let’s find the coordinates of the points where both graphs intersect, for this we solve the following equation:

2 * x – x² = 0,

x * (2 – x) = 0, whence we get x = 0 and x = 2.

Since the coefficient at x² is positive, we are looking for the area of the parabolic segment over the Ox axis, for this we calculate the integral:

s = integral (0 to 2) (2 * x – x²) dx,

s = x² – x³ / 3 (from 0 to 2),

s = 4 – 8/3 = 4/3 units ².

Answer: the area of the segment is 4/3 units ².