Question

40 баллов! Обчислити площу фігури обмеженої лініями y=-x²-4x, y=4+x.Тема: інтеграли

Answer 1

См фото
=============================

Answer 2

Calculation of the Area Bounded by the Curves y = -x² - 4x and y = 4 + x

To calculate the area bounded by the curves y = -x² - 4x and y = 4 + x, we need to find the points of intersection between the two curves and then integrate the difference between the two curves over that interval.

Let's start by finding the points of intersection between the two curves.

To find the points of intersection, we can set the two equations equal to each other:

-x² - 4x = 4 + x

Rearranging the equation, we get:

x² + 5x + 4 = 0

Now, we can solve this quadratic equation to find the values of x.

Using the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), where a = 1, b = 5, and c = 4, we can calculate the values of x.

Calculating the discriminant, b² - 4ac, we get:

(5)² - 4(1)(4) = 25 - 16 = 9

Since the discriminant is positive, we have two real solutions for x.

Using the quadratic formula, we get:

x = (-5 ± √9) / 2

Simplifying further, we have:

x₁ = (-5 + 3) / 2 = -1 x₂ = (-5 - 3) / 2 = -4

So, the points of intersection between the two curves are (-1, 3) and (-4, 0).

Now that we have the points of intersection, we can integrate the difference between the two curves over the interval from x = -4 to x = -1 to find the area bounded by the curves.

The formula for calculating the area between two curves is:

A = ∫[a,b] (f(x) - g(x)) dx

where f(x) is the upper curve and g(x) is the lower curve.

In this case, the upper curve is y = 4 + x and the lower curve is y = -x² - 4x.

So, the area A is given by:

A = ∫[-4,-1] ((4 + x) - (-x² - 4x)) dx

Simplifying the equation, we have:

A = ∫[-4,-1] (x² + 5x + 4) dx

Now, we can integrate the equation to find the area.

Integrating the equation, we get:

A = [x³/3 + (5x²)/2 + 4x] from -4 to -1

Substituting the limits of integration, we have:

A = [(-1)³/3 + (5(-1)²)/2 + 4(-1)] - [(-4)³/3 + (5(-4)²)/2 + 4(-4)]

Simplifying further, we get:

A = [(-1/3) + (5/2) - 4] - [(-64/3) + (80/2) - 16]

A = [(-1/3) + (5/2) - 4] - [(-64/3) + 40 - 16]

A = [(-1/3) + (5/2) - 4] - [(-64/3) + 40 - 16]

A = [-1/3 + 15/6 - 12/3] - [-64/3 + 120/3 - 48/3]

A = [-1/3 + 15/6 - 12/3] - [-64/3 + 120/3 - 48/3]

A = [-2/6 + 15/6 - 24/6] - [-64/3 + 120/3 - 48/3]

A = [-11/6] - [8/3]

A = -11/6 - 8/3

A = -11/6 - 16/6

A = -27/6

Simplifying further, we get:

A = -9/2

Therefore, the area bounded by the curves y = -x² - 4x and y = 4 + x is -9/2 square units.