Question

Помогите взавтра у меня екзамен:::::: Обчислить плошу фигури обмеженой параболи у=х-3х+4 и прямою у=4-х ребята я вас просю помогите мне сейчас

Answer 1

Надо вычислить площадь фигуры, заштрихованной на чертеже.

Парабола y = х^2 - 3*х + 4; прямая y = 4 - x;

пересекаются в точках x = 0 и 2 (просто приравниваем х^2 - 3*х + 4 = 4 - х, и сразу это получаем).

Дальше решние на втором рисунке. Площадь 4/3

Answer 2

Calculating the Area of a Figure Bounded by a Parabola and a Line

To calculate the area of a figure bounded by a parabola and a line, we need to find the points of intersection between the parabola and the line. Once we have these points, we can determine the limits of integration and set up the integral to calculate the area.

Let's start by finding the points of intersection between the parabola and the line.

The given equations are: - Parabola: y = x^2 - 3x + 4 - Line: y = 4 - x

To find the points of intersection, we need to solve the system of equations formed by equating the two equations:

x^2 - 3x + 4 = 4 - x

Simplifying the equation, we get:

x^2 - 2x = 0

Factoring out x, we have:

x(x - 2) = 0

This equation has two solutions: x = 0 and x = 2.

Now that we have the x-coordinates of the points of intersection, we can find the corresponding y-coordinates by substituting these values into either of the original equations.

For x = 0: y = 0^2 - 3(0) + 4 = 4

For x = 2: y = 2^2 - 3(2) + 4 = 2

Therefore, the points of intersection are (0, 4) and (2, 2).

To calculate the area of the figure bounded by the parabola and the line, we can set up the integral as follows:

∫[a, b] (f(x) - g(x)) dx

where f(x) is the upper function (in this case, the parabola) and g(x) is the lower function (in this case, the line). The limits of integration, a and b, are the x-coordinates of the points of intersection.

In this case, the integral becomes:

∫[0, 2] ((x^2 - 3x + 4) - (4 - x)) dx

Simplifying the integral, we get:

∫[0, 2] (x^2 - 2x) dx

Integrating term by term, we have:

∫[0, 2] (x^2 - 2x) dx = [(1/3)x^3 - x^2] evaluated from 0 to 2