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The problem asks to compute the area of the shaded region under the line \( y = x + 1 \) between \( x = 0 \) and \( x = 2 \).

### Step-by-step solution:

1. **Identify the equation of the line:**
   The line given is \( y = x + 1 \).
   
2. **Determine the limits of integration:**
   The shaded area lies between \( x = 0 \) and \( x = 2 \) on the x-axis.

3. **Set up the integral:**
   To find the area under the line, we integrate the function \( y = x + 1 \) from \( x = 0 \) to \( x = 2 \).

   \[
   A = \int_{0}^{2} (x + 1) \, dx
   \]

4. **Compute the integral:**
   Now, solve the integral step by step:
   \[
   A = \int_{0}^{2} (x + 1) \, dx = \left[ \frac{x^2}{2} + x \right]_0^2
   \]
   Substituting the limits:
   \[
   A = \left( \frac{2^2}{2} + 2 \right) - \left( \frac{0^2}{2} + 0 \right)
   \]
   Simplifying:
   \[
   A = \left( \frac{4}{2} + 2 \right) = 2 + 2 = 4
   \]

### Final Answer:
The area of the shaded region is **4 square units**.

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Would you like a more detailed explanation or have any questions?

Here are five related questions for practice:
1. What would be the area if the limits of integration were from \( x = 1 \) to \( x = 3 \)?
2. How does the area change if the line equation is \( y = 2x + 1 \)?
3. What is the area under the curve \( y = x^2 + 1 \) between \( x = 0 \) and \( x = 2 \)?
4. How do you compute the area between two curves, such as \( y = x+1 \) and \( y = x^2 \)?
5. What is the significance of negative areas in definite integrals?

**Tip:** When finding the area under a curve, always pay attention to the limits of integration and whether the curve crosses the x-axis.

Обчислити площу заштрихованої фігури (Compute the area of the shaded region).

Learn how to compute the area of the shaded region under the line y = x + 1 between x = 0 and x = 2 using integration techniques. This problem covers key concepts in calculus, such as integration and the Fundamental Theorem of Calculus. Suitable for students in Grades 10-12.

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