Sketch the graph y  x  2 . Evaluate  x  2dx . What does the value of the integral represent on the
0
graph?

To address your request, let's break it down into three parts:

1. **Sketch the Graph \( y = x - 2 \):**

   The equation \( y = x - 2 \) represents a linear function with a slope of 1 and a y-intercept at \( -2 \). This means the line crosses the y-axis at \( (0, -2) \) and has a positive slope, increasing by 1 unit in the y-direction for every 1 unit increase in the x-direction.

2. **Evaluate \( \int (x - 2) \, dx \):**

   To evaluate the indefinite integral \( \int (x - 2) \, dx \):

   \[
   \int (x - 2) \, dx = \int x \, dx - \int 2 \, dx
   \]

   \[
   = \frac{x^2}{2} - 2x + C
   \]

   Here, \( C \) is the constant of integration.

3. **Interpretation of the Integral on the Graph:**

   The value of the definite integral of \( x - 2 \) from 0 to some value \( a \) (let's say) represents the **area under the curve** \( y = x - 2 \) from \( x = 0 \) to \( x = a \).

   If we were to compute the definite integral from 0 to \( a \):

   \[
   \int_{0}^{a} (x - 2) \, dx = \left[\frac{x^2}{2} - 2x\right]_{0}^{a}
   \]

   Substituting the limits:

   \[
   = \left(\frac{a^2}{2} - 2a\right) - \left(\frac{0^2}{2} - 2 \times 0\right)
   \]

   \[
   = \frac{a^2}{2} - 2a
   \]

   This result represents the net area between the graph of \( y = x - 2 \) and the x-axis over the interval from \( x = 0 \) to \( x = a \). Depending on \( a \), this area might be positive, negative, or zero. The area above the x-axis contributes positively, while the area below the x-axis contributes negatively.

If you have any questions or need more details, let me know! Here are 5 questions that expand on this information:

1. What happens to the integral when the upper limit \( a \) is less than 2?
2. How would the graph of \( y = x - 2 \) look if the slope changed?
3. What is the physical interpretation of the area under the curve for this specific function?
4. How does the value of the constant \( C \) affect the indefinite integral?
5. What are the implications of finding the area under the curve from \( x = 2 \) to \( x = 4 \)?

**Tip:** When evaluating definite integrals, always consider the areas above and below the x-axis separately, as they may cancel each other out!

Learn how to sketch the graph of y = x - 2 and evaluate the integral ∫(x - 2) dx step-by-step. Understand the interpretation of the integral as the area under the curve and its significance on the graph. Suitable for students in Grades 11-12.

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