Question

Вычислить площадь фигуры, ограниченной линиями y=x^2+x+1, y=x+2. Нарисуйте графики.

Answer 1

смотрите фото.....................

Answer 2

Calculating the Area of the Figure

To calculate the area of the figure bounded by the lines y=x^2+x+1 and y=x+2, we need to find the points of intersection between these two curves. Once we have the points of intersection, we can integrate the difference between the two curves to find the area.

First, let's find the points of intersection by setting the two equations equal to each other and solving for x:

y=x^2+x+1 y=x+2

By setting these two equations equal to each other, we can solve for x:

x^2 + x + 1 = x + 2 x^2 = 1 x = ±1

So, the points of intersection are at x = 1 and x = -1.

Now, we can integrate the difference between the two curves from x = -1 to x = 1 to find the area of the figure.

Graphical Representation

Let's plot the graphs of the two equations to visualize the figure.

The graph of y=x^2+x+1 is a parabola, and the graph of y=x+2 is a straight line. We can use these equations to plot the graphs.

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-2, 2, 400)
y1 = x**2 + x + 1
y2 = x + 2

plt.plot(x, y1, label='y=x^2+x+1')
plt.plot(x, y2, label='y=x+2')
plt.fill_between(x, y1, y2, where=(y1 > y2), interpolate=True, color='gray', alpha=0.3)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graphs of y=x^2+x+1 and y=x+2')
plt.legend()
plt.grid(True)
plt.show()

The shaded region between the two curves represents the area of the figure bounded by the lines y=x^2+x+1 and y=x+2.

This visualization helps us understand the geometric representation of the area enclosed by the given curves.