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You've uploaded a mathematical worksheet containing several problems related to finding areas under curves and areas enclosed by geometric shapes. Let's go through the questions one by one, providing step-by-step solutions.

### 1. **Hallar el área limitada por la recta \( x + y = 10 \), el eje \( OX \), y las ordenadas de \( x = 2 \) y \( x = 8 \).**

To solve this, we need to calculate the area under the line \( x + y = 10 \) between the points where \( x = 2 \) and \( x = 8 \).

- Rearrange the equation: \( y = 10 - x \).
- The area under the curve is given by the integral:
  \[
  \text{Área} = \int_{2}^{8} (10 - x) \, dx
  \]
- Solving this integral gives us the area.

### 2. **Calcular el área del recinto limitado por la curva \( y = 9 - x^2 \) y el eje OX.**

The curve \( y = 9 - x^2 \) is a parabola opening downwards. We want the area between the parabola and the x-axis.

- Find the points where the curve intersects the x-axis by setting \( y = 0 \):
  \[
  9 - x^2 = 0 \quad \Rightarrow \quad x = \pm 3
  \]
- The area is given by:
  \[
  \text{Área} = \int_{-3}^{3} (9 - x^2) \, dx
  \]
- Solving this integral will give the required area.

### 3. **Calcular el área del triángulo de vértices A(3, 0), B(6, 3), C(8, 0).**

We can use the formula for the area of a triangle given three vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \):
\[
\text{Área} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting \( A(3, 0) \), \( B(6, 3) \), and \( C(8, 0) \) into the formula will give the area.

### 4. **Calcular el área limitada por las gráficas de las funciones \( y^2 = 4x \) y \( y = x^2 \).**

These curves are a parabola \( y^2 = 4x \) (a sideways parabola) and \( y = x^2 \) (a standard upward parabola). We need to find the points of intersection and then set up the integral:

- Find the points of intersection by solving \( y^2 = 4x \) and \( y = x^2 \).
- Set up the integral for the area between these curves.

### 5. **Calcular el área limitada por la curva \( xy = 36 \), el eje OX y las rectas \( x = 6 \) y \( x = 12 \).**

The curve is a hyperbola. The limits of integration are \( x = 6 \) and \( x = 12 \), and we need to integrate the function \( y = \frac{36}{x} \) between these values:

\[
\text{Área} = \int_{6}^{12} \frac{36}{x} \, dx
\]

### 6. **Calcular el área limitada por la curva \( y = 2(1 - x^2) \) y la recta \( y = -1 \).**

We need to calculate the area between the curve \( y = 2(1 - x^2) \) and the line \( y = -1 \).

- Set the two equations equal to each other to find the limits of integration.
- The area is given by integrating the difference between the two functions.

### 7. **Calcular el área del recinto limitado por la parábola \( y = x^2 + 2 \) y la recta que pasa por los puntos \( (-1, 0) \) y \( (1, 4) \).**

First, find the equation of the line passing through \( (-1, 0) \) and \( (1, 4) \), then set up the integral for the area between the parabola and the line.

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### 5 Related Questions:
1. How do you find the limits of integration for two intersecting curves?
2. What is the geometric interpretation of an integral?
3. How do you determine the equation of a line given two points?
4. What steps are involved in solving an integral for a parabolic area?
5. How can you use symmetry to simplify integrals of symmetric functions?

### Tip:
Always sketch the graphs of the curves involved to visualize the regions you're integrating over. This helps in setting the correct limits and understanding the area you're calculating.

Find the areas under curves and enclosed by geometric shapes for the given mathematical problems.

Learn how to calculate the area under curves and between graphs for various mathematical problems using calculus and definite integrals. Detailed solutions cover parabola, triangle area calculations, and integration between functions. Suitable for Grade 11 math students.

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