Question

Y=5-x^2 y=1 найдите площадь фигуры, ограниченной линией

Answer 1

1=5-x^2
x^2=5-1
x^2=4
x=2

Answer 2

Calculating the Area of the Figure

To find the area of the figure bounded by the given lines, we can use the definite integral to calculate the area under the curve. The given lines are y = 5 - x^2 and y = 1.

The area can be calculated using the formula: S = ∫[a, b] (f(x) - g(x)) dx

Where: - f(x) is the upper function (in this case, y = 5 - x^2) - g(x) is the lower function (in this case, y = 1) - a and b are the x-values where the two functions intersect

Calculating the Intersection Points

To find the intersection points, we need to solve the system of equations formed by the two given lines: y = 5 - x^2 and y = 1.

By setting the two equations equal to each other, we can find the x-values where the lines intersect.

5 - x^2 = 1

Solving for x: x^2 = 4 x = ±2

So, the intersection points are at x = -2 and x = 2.

Calculating the Area

Now, we can calculate the area using the definite integral: S = ∫[-2, 2] (5 - x^2 - 1) dx

This simplifies to: S = ∫[-2, 2] (4 - x^2) dx

Integrating with respect to x: S = [4x - (x^3)/3] |[-2, 2]

Substituting the limits of integration: S = [8 - (8/3)] - [-8 + (8/3)] S = (24/3) - (-24/3) S = 48/3 S = 16

So, the area of the figure bounded by the given lines is 16 square units.