Question

Найти площади фигуры, ограниченной графиками функций y=x^2+2 и y=x+4

Answer 1

Ответ:

Площадь фигуры равняется 4,5

Объяснение:

1) Найдём точки пересечения графиков

x²+2 = x+4
x²-x+2-4 = 0
x²-x-2 = 0
D = (-1)²-4*1*(-2) = 1+8 = 9 = 3²
x₁₂ = (1±3)/(2*1)
x₁ = 2 ; x₂ = -1

2) Найдём какой график проходит над другим, например в точке х=0

a) у = 0²+2
у = 2
б) у = 0+4
у = 0

Вторая функция проходит над первой

3) Находим площадь

$\displaystyle \int\limits^2_{-1} {(x+4-(x^2+2))} \, dx =\int\limits^2_{-1} {(x+4-x^2-2)} \, dx = \int\limits^2_{-1} {(-x^2+x+2)} \, dx =-\frac{x^3}{3}+\frac{x^2}{2}+2x|^{2}_{-1} =(-\frac{2^3}{3} +\frac{2^2}{2}+2*2)-(-\frac{(-1)^3}{3} +\frac{(-1)^2}{2}+2*(-1))=(-\frac{8}{3} +\frac{4}{2}+4)-(-\frac{-1}{3}+\frac{1}{2}-2) =-\frac{8}{3}+2+4-\frac{1}{3}-\frac{1}{2}+2=\frac{-8-1}{3}+8-0,5=-3+7,5=4,5$

Answer 2

Finding the Area of the Region Bounded by the Graphs of the Functions y=x^2+2 and y=x+4

To find the area of the region bounded by the graphs of the functions y=x^2+2 and y=x+4, we need to determine the points of intersection between the two functions and then calculate the definite integral of the absolute difference between the two functions over the interval of intersection.

1. Finding the Points of Intersection: - To find the points of intersection, we set the two functions equal to each other and solve for x. - The points of intersection will be the x-values where the two functions are equal.

2. Calculating the Definite Integral: - Once we have the points of intersection, we can calculate the definite integral of the absolute difference between the two functions over the interval of intersection. - The definite integral will give us the area of the region bounded by the two functions.

Solution Steps:

1. Finding the Points of Intersection: - Setting the two functions equal to each other: - x^2+2 = x+4 - Solving for x: - x^2 - x + 2 = 0 - Using the quadratic formula to find the solutions for x.

2. Calculating the Definite Integral: - Once we have the points of intersection, we can calculate the definite integral of the absolute difference between the two functions over the interval of intersection.

Conclusion:

By following these steps, we can find the area of the region bounded by the graphs of the functions y=x^2+2 and y=x+4.