Calculate the area of the shape bounded by lines y = 3x ^ 2, y = 0, x = -3, x = 2

Since the parabola y = 3x ^ 2 has a common point x0 = 0 with the oX axis, the area of the figure S formed by the given lines will be equal to the sum of the integrals:

S = ∫3x ^ 2 * dx | -3; 0 + ∫3x * dx | 0; 2 = x ^ 3 | -3; 0 + x ^ 3 | 0; 1 = (0 – (-3) ^ 3) + (1 ^ 3 – 0) = 27 + 1 = 28.

Answer: the required area is 28.