Calculate the area of a shape bounded by lines y ^ 2 = x ^ 3; 0≤x≤4 / 3

Let us raise the equation of the function to the 1/2 power:

y = x ^ (3/2).

Then the area of the figure S formed by the given lines will be equal to the integral:

S = ∫x ^ (3/2) * dx | 0; 4/3 = 2/5 * x ^ (5/2) | 0; 4/3 = 2/5 * (4/3) ^ (5/2) = 2/5 * (2 / √3) ^ 5 = 64/5 (√3) ^ 5.

Answer: the required area is 64/5 (√3) ^ 5.