Calculate the area of the shape bounded by the lines y = x ^ 2 y = x ^ -1 y = e.

Let’s find the point of intersection of the graphs of functions, for this we equate their equations to each other:

x ^ 2 = 1 / x;

x ^ 3 = 1;

x = 1.

Calculate the point of intersection of the line y = x ^ 2 with the oX axis:

x ^ 2 = 0;

x = 0.

Then the area S of the figure formed by the lines is equal to the sum of the integrals:

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

Answer: the required area is 4/3.