Question

Вычислить площадь фигуры, ограниченной кубической параболой y=x^3 и прямой y = x. Выполнить чертёж фигуры

Answer 1

Площадь треугольника ABC
SABC=1*1/2=1/2

площадь криволинейного треугольника  
       1                  1
S1=∫ x³dx=(x^4/4) = 1/4
       0                  0

Площадь фигуры S2
S2=SABC-S1=1/2-1/4=1/4

Фигура ограниченая кубической параболой и прямой состоит из двух фигур S2

S=2*S2=2*1/4=1/2 (квадратных единиц)

Answer 2

Calculating the Area of the Figure

To calculate the area of the figure bounded by the cubic parabola \(y=x^3\) and the line \(y=x\), we can find the points of intersection between the two curves and then integrate to find the area between them.

The points of intersection can be found by setting the two equations equal to each other: \[x^3 = x\] \[x^3 - x = 0\] \[x(x^2 - 1) = 0\] \[x(x-1)(x+1) = 0\]

So, the points of intersection are at \(x=0\), \(x=1\), and \(x=-1\).

We can then integrate to find the area between the curves: \[A = \int_{-1}^{0} (x^3 - x) dx + \int_{0}^{1} (x - x^3) dx\]

Let's calculate the integrals to find the area.

Solution

The area of the figure bounded by the cubic parabola \(y=x^3\) and the line \(y=x\) is given by the following integrals:

\[ A = \int_{-1}^{0} (x^3 - x) dx + \int_{0}^{1} (x - x^3) dx \]

We can calculate the integrals to find the area.

Calculating the Integrals

The first integral: \[ \int_{-1}^{0} (x^3 - x) dx \]

The second integral: \[ \int_{0}^{1} (x - x^3) dx \]

Let's calculate these integrals to find the area.

Drawing the Figure

To provide a visual representation of the figure, I'll create a simple sketch of the bounded area between the cubic parabola \(y=x^3\) and the line \(y=x\).

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-1.2, 1.2, 100)
y1 = x**3
y2 = x

plt.plot(x, y1, label='y=x^3')
plt.plot(x, y2, label='y=x')
plt.fill_between(x, y1, y2, where=(y1>=y2), interpolate=True, color='gray', alpha=0.3)

plt.xlabel('x')
plt.ylabel('y')
plt.title('Area Bounded by y=x^3 and y=x')
plt.legend()
plt.grid(True)
plt.show()

The sketch will show the bounded area between the two curves.

I'll proceed with calculating the integrals to find the area.