The area of the figure bounded by the lines y = x ^ 2, y = 1 is equal to.

1. The area of the figure, bounded by the lines y = x ^ 2 – from below, and y = 1 – from above, is equal to a definite integral of the difference of these functions f (x) = 1 – x ^ 2 in the range from x = x1 to x = x2, where x1 and x2 are the abscissas of the intersection points of two lines:

x ^ 2 = 1;
x = ± 1;
x1 = -1;
x2 = 1.
2. Let’s calculate the integral of the function f (x):

F (x) = ∫f (x) dx = ∫ (1 – x ^ 2) dx = x – x ^ 3/3;
F (x1) = F (-1) = -1 – (-1) ^ 3/3 = -1 + 1/3 = -2/3;
F (x2) = F (1) = 1 – 1 ^ 3/3 = 1 – 1/3 = 2/3;
S = F (x2) – F (x1) = 2/3 – (-2/3) = 2/3 + 2/3 = 4/3.
Answer: S = 4/3.