Use the limit definition to find the area of the region bounded by the graph of the following 
function 𝑓(𝑥) = 3𝑥 − 2 on [2,5]

To find the area of the region bounded by the graph of the function \( f(x) = 3x - 2 \) over the interval \([2, 5]\), we can use the limit definition of a definite integral. The limit definition expresses the integral as the limit of Riemann sums, which can be used to find the area under the curve.

### Step-by-Step Solution:

1. **Partition the interval** \([2, 5]\) into \(n\) subintervals of equal width:
   \[
   \Delta x = \frac{b - a}{n} = \frac{5 - 2}{n} = \frac{3}{n}
   \]
   where \( a = 2 \) and \( b = 5 \).

2. **Choose sample points** \( x_i^* \) in each subinterval. Typically, the right endpoints are chosen, so the sample points are:
   \[
   x_i^* = a + i\Delta x = 2 + i \cdot \frac{3}{n}
   \]
   for \( i = 1, 2, \dots, n \).

3. **Evaluate the function** at each sample point:
   \[
   f(x_i^*) = 3(2 + i \cdot \frac{3}{n}) - 2 = 6 + 9\frac{i}{n} - 2 = 4 + \frac{9i}{n}
   \]

4. **Write the Riemann sum** for the area:
   \[
   A_n = \sum_{i=1}^{n} f(x_i^*) \Delta x = \sum_{i=1}^{n} \left( 4 + \frac{9i}{n} \right) \cdot \frac{3}{n}
   \]

5. **Simplify the sum**:
   \[
   A_n = \frac{3}{n} \sum_{i=1}^{n} \left( 4 + \frac{9i}{n} \right)
   = \frac{3}{n} \left( \sum_{i=1}^{n} 4 + \sum_{i=1}^{n} \frac{9i}{n} \right)
   \]

   Using summation formulas:
   \[
   \sum_{i=1}^{n} 1 = n, \quad \sum_{i=1}^{n} i = \frac{n(n+1)}{2}
   \]

   We can simplify the sums:
   \[
   A_n = \frac{3}{n} \left( 4n + \frac{9}{n} \cdot \frac{n(n+1)}{2} \right)
   = \frac{3}{n} \left( 4n + \frac{9n(n+1)}{2n} \right)
   = 3 \left( 4 + \frac{9(n+1)}{2n} \right)
   \]

6. **Take the limit** as \( n \to \infty \):
   \[
   \lim_{n \to \infty} A_n = 3 \lim_{n \to \infty} \left( 4 + \frac{9(n+1)}{2n} \right)
   = 3 \left( 4 + \frac{9}{2} \right)
   = 3 \cdot \frac{17}{2} = \frac{51}{2} = 25.5
   \]

Thus, the area of the region bounded by the graph of \( f(x) = 3x - 2 \) on \([2, 5]\) is \( \boxed{25.5} \).

---

Would you like more details on any part of this process?

Here are some additional questions to explore:
1. How does changing the interval affect the result of the area calculation?
2. Can you generalize the limit definition for other linear functions?
3. What happens if we choose left endpoints instead of right endpoints for the Riemann sum?
4. How does the summation formula simplify the calculation for general functions?
5. How does this limit definition relate to the Fundamental Theorem of Calculus?

**Tip:** The limit definition of an integral is especially helpful for understanding the concept of area under a curve as the sum of small rectangles.

Use the limit definition to find the area of the region bounded by the graph of the following function 𝑓(𝑥) = 3𝑥 − 2 on [2,5]

Discover how to calculate the area of the region bounded by the graph of the linear function f(x) = 3x - 2 over the interval [2,5] using the limit definition of the definite integral. This step-by-step solution covers Riemann sums, summation formulas, and calculus concepts. Suitable for Grades 11-12.

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