Calculate the area of the figure bounded by the lines: y = x² + 4x + 4; y = x + 4.

Find the intersection points of the lines, for this we equate their equations to each other:

x ^ 2 + 4x + 4 = x + 4;

x ^ 2 + 3x = 0;

x * (x + 3) = 0;

x1 = 0; x2 = -3.

The area bounded by the lines is equal to the difference of the integrals:

S = ∫ (x ^ 2 + 4x + 4) * dx | -3; 0 – ∫ (x + 4) * dx | -3; 0 = (1 / 3x ^ 3 + 2x ^ 2 + 4x) | -3 ; 0 – (1 / 2x ^ 2 + 4x) | -3; 0 = 1/3 * (-3) ^ 3 + 3/2 * (-3) ^ 2 = -9 + 27/2 = 9/2 .