Calculate the area of the figure bounded by the straight lines x = 2 and x = 3, the parabola y = -x ^ 2 + 6x-5 and the Ox axis.

Calculate the area of a figure bounded by straight lines and a parabola:

x = 2;

x = 3;

y = -x ^ 2 + 6 * x – 5;

S = (2 to 3) ∫ (-x ^ 2 + 6 * x – 5) dx = (2 to 3) (∫-x ^ 2 dx + 6 ∫x dx – 5 ∫dx) = (from 2 up to 3) (-x ^ 3/3 + 6 * x ^ 2/2 – 5 * x = (2 to 3) (-1/3 * x ^ 3 + 6/2 * x ^ 2 – 5 * x ) = (2 to 3) (-1/3 * x ^ 3 + 3 * x ^ 2 – 5 * x) = -1/3 * 3 ^ 3 + 3 * 3 ^ 2 – 5 * 3 – (- 1/3 * 2 ^ 2 + 3 * 2 ^ 2 – 5 * 2) = -9 + 27 – 15 – (-4/3 + 12 – 10) = 13 – 9 – (-4/3 + 2) = 4 + 4/3 – 2 = 2 + 4/3 = 2 + 1 + 1/3 = 3 + 1/3 = 3 1/3 = 3.66.

Answer: S = 3.66.