Question

Помогите найти площадь фигуры ограниченной линиями y=x^3 и y=x

Answer 1

$\int\limits^1_0 {(x-x^3)} \, dx = \frac{x^2}{2}- \frac{x^4}{4}/^1_0= \\ \\ = \frac{1}{2}- \frac{1}{4}= 0.5-0.25=0.25$

Answer 2

Finding the Area of the Region Bounded by the Curves y=x^3 and y=x

To find the area of the region bounded by the curves y=x^3 and y=x, we can use the definite integral of the absolute difference between the two functions over the interval where they intersect.

The points of intersection can be found by setting the two functions equal to each other and solving for x:

x^3 = x x(x^2 - 1) = 0 x(x-1)(x+1) = 0 So, the points of intersection are x = -1, x = 0, and x = 1.

The area A of the region bounded by the curves y=x^3 and y=x can be found using the definite integral:

A = ∫[a, b] |f(x) - g(x)| dx

Where f(x) = x^3 and g(x) = x, and a and b are the x-coordinates of the points of intersection.

Now, let's calculate the area using the definite integral:

A = ∫[-1, 0] (x - x^3) dx + ∫[0, 1] (x^3 - x) dx

We can solve these integrals to find the area of the region bounded by the curves y=x^3 and y=x.