Calculate the area of the shape bounded by the lines y ^ 3 = x, y = 1, x = 8.

Let’s transform the cubic function:

y³ = x,

y = ³√x.

Based on the construction, it is required to calculate the area bounded by a cubic parabola and straight lines y = 1 and x = 8, located in the 1st coordinate quarter. The required area is expressed by the integral:

s = integral (1 to 8) (³√x – 1) dx = 3 * ³√ (x ^ 4) / 4 – x (1 to 8) = 12 – 8 – 3/4 + 1 = 5 – 3/4 = 17/4 units ².

Answer: the area of the bounded figure is 17/4 units ².