Calculate the area of the shape bounded by the lines y = x ^ 2-4x + 5 and y = 5.

Equating the equations of the lines to each other, we find the intersection points of the graphs:

x ^ 2 – 4x + 5 = 5

x * (x – 4) = 0

x1 = 0; x2 = 4

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

∫5 * dx | 0; 4 – ∫ (x ^ 2 – 4x + 5) * dx | 0; 4 = 5 * 4 – (1/3 * 8 – 2 * 4 + 5 * 4) = -8/3 + 8 = 16/3.