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Te njehsohet syprina e siperfaqes se figures se kufizuar me parabolen y=x^2-2x dhe drejtezen : x=-3

The problem is asking to find the area of the surface bounded by the parabola \( y = x^2 - 2x \) and the line \( x = -3 \).

To find the area, we need to perform the following steps:

1. **Determine the points of intersection**: Find the points where the parabola intersects the line \( x = -3 \).
2. **Find the limits of integration**: These are typically the x-values that define the region of interest.
3. **Set up the integral**: The area under the curve of the parabola from \( x = -3 \) to the other x-value of interest.
4. **Calculate the integral**: This will give the area under the parabola between the given bounds.

### Step-by-Step Solution

1. **Find the intersection points**: The line \( x = -3 \) is a vertical line, so we substitute \( x = -3 \) into the parabola equation to find the corresponding y-value.
   \[
   y = (-3)^2 - 2(-3) = 9 + 6 = 15
   \]
   So, the point of intersection is \( (-3, 15) \).

2. **Determine the second boundary**: The problem does not specify another boundary, but typically, we look for where the parabola intersects the x-axis. Let's find those points by setting \( y = 0 \):
   \[
   x^2 - 2x = 0 \\
   x(x - 2) = 0
   \]
   So, the roots are \( x = 0 \) and \( x = 2 \).

3. **Set up the integral**: The area is found by integrating the parabola from \( x = -3 \) to \( x = 0 \). This is because the vertical line \( x = -3 \) defines one boundary, and the parabola intersects the x-axis at \( x = 0 \).

   The integral is:
   \[
   \text{Area} = \int_{-3}^{0} (x^2 - 2x) \, dx
   \]

4. **Calculate the integral**:

   \[
   \int (x^2 - 2x) \, dx = \frac{x^3}{3} - x^2
   \]

   Evaluating from \( -3 \) to \( 0 \):
   \[
   \left[\frac{x^3}{3} - x^2\right]_{-3}^{0} = \left(0 - 0\right) - \left(\frac{-27}{3} - 9\right) = 0 - (-18) = 18
   \]

So, the area of the surface is \( 18 \) square units.

Would you like more details or have any questions? 

### Related Questions:
1. How do we find the intersection points of two general curves?
2. What are the properties of a parabola and how do they help in area calculations?
3. How do we handle integrals when the region is below the x-axis?
4. What is the geometrical significance of the definite integral?
5. How can we find the area between two curves?

**Tip:** Always sketch the graph of the functions involved in the problem to get a visual understanding of the bounded area.

Të njihet syprina e sipërfaqes së figurës së kufizuar me parabolën y = x^2 - 2x dhe drejtëzën x = -3.

Learn how to calculate the area of the surface bounded by the parabola y = x^2 - 2x and the line x = -3 using definite integrals. Step-by-step solution involving calculus concepts suitable for students in Grades 11-12.

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