Question

Найти площадь фигуры ограниченной параболами y=(x-2)^2, y=(x+2)^2, прямой y=1 и осью Оx Нужно не только решение (по формуле), но и график и желательно подробно расписать как вы этот график фигуры построили

Answer 1

1)  Построим графики у=(х-2)^2 и у=(х+2)^2
а)  у=(х-2)^2=x^2-4x+4 (график - парабола, ветви вверх)
1. Найдем точки пересечения с осью Ох
x^2-4x+4=0;   D=16-16=0;   х=2
2. Вершина имеет координаты (2;0)
3. Пересекается с осью Оу в точке (0;4)
4. Построим график (см. рисунок)
б)  у=(х+2)^2=x^2+4x+4 (график - парабола, ветви вверх)
1. Найдем точки пересечения с осью Ох
x^2+4x+4=0;   D=16-16=0;   х=-2
2. Вершина имеет координаты (-2;0)
3. Пересекается с осью Оу в точке (0;4)
4. Построим график (см. рисунок)
в) Проведем прямую у=1
2) Найдем площадь фигуры ограниченной параболами и прямой у=1 (заштрихована на рисунке)
Площадь найдете как сумма трех интегралов

Answer 2

Finding the Area of the Figure Bounded by Parabolas and a Line

To find the area of the figure bounded by the parabolas y = (x-2)^2, y = (x+2)^2, the line y = 1, and the x-axis, we can follow these steps:

1. Graphing the Equations: - First, let's graph the equations to visualize the figure. Here is the graph:


2. Identifying the Intersection Points: - To find the area, we need to determine the x-values where the curves intersect. These points will be the limits of integration. - The parabolas intersect at x = 0 and x = 4. - The line y = 1 intersects the parabolas at x = -3 and x = 3.

3. Setting Up the Integral: - Since the figure is symmetric, we can calculate the area of one side and then multiply it by 2 to get the total area. - The integral to find the area of one side is given by: A = 2 * ∫[a, b] (f(x) - g(x)) dx, where f(x) and g(x) are the upper and lower curves, respectively, and a and b are the limits of integration. - In this case, the upper curve is y = (x+2)^2 and the lower curve is y = 1. - The limits of integration for one side are x = -3 to x = 0.

4. Evaluating the Integral: - Now, let's evaluate the integral to find the area of one side. - The integral becomes: A = 2 * ∫[-3, 0] ((x+2)^2 - 1) dx. - Simplifying the integrand, we get: A = 2 * ∫[-3, 0] (x^2 + 4x + 3) dx. - Evaluating the integral, we find: A = 2 * [(1/3)x^3 + 2x^2 + 3x] |[-3, 0]. - Plugging in the limits of integration, we get: A = 2 * [(1/3)(0)^3 + 2(0)^2 + 3(0)] - [(1/3)(-3)^3 + 2(-3)^2 + 3(-3)]. - Simplifying further, we find: A = 2 * [0 - (-9)] = 18.

5. Calculating the Total Area: - Since the figure is symmetric, the total area is twice the area of one side. - Therefore, the total area of the figure bounded by the parabolas, the line, and the x-axis is 2 * 18 = 36 square units.

Conclusion

The area of the figure bounded by the parabolas y = (x-2)^2, y = (x+2)^2, the line y = 1, and the x-axis is 36 square units. This was calculated by evaluating the integral of the difference between the upper and lower curves over the appropriate limits of integration.