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

To find the area of the region bounded by the graph of the function \( f(x) = 3x - 2 \) on the interval \([2,5]\), we will use the limit definition of a definite integral.

The limit definition of the area under a curve is given by:

\[
\text{Area} = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x
\]

Where:
- \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval,
- \( x_i^* \) is a sample point in each subinterval \([x_{i-1}, x_i]\),
- \( f(x_i^*) \) is the function value at the sample point,
- \( a \) and \( b \) are the endpoints of the interval.

For the given problem:
- \( f(x) = 3x - 2 \),
- \( a = 2 \), \( b = 5 \).

### Step 1: Set up the partition
\[
\Delta x = \frac{5 - 2}{n} = \frac{3}{n}
\]

We will choose \( x_i^* \) as the right endpoint of each subinterval:
\[
x_i^* = a + i\Delta x = 2 + i\frac{3}{n}
\]

### Step 2: Calculate the sum
The area under the curve is approximated by the sum:
\[
\sum_{i=1}^{n} f(x_i^*) \Delta x = \sum_{i=1}^{n} \left( 3(2 + i\frac{3}{n}) - 2 \right) \cdot \frac{3}{n}
\]
Simplifying \( f(x_i^*) \):
\[
f(x_i^*) = 3(2 + i\frac{3}{n}) - 2 = 6 + \frac{9i}{n} - 2 = 4 + \frac{9i}{n}
\]

Thus, the sum becomes:
\[
\sum_{i=1}^{n} \left( 4 + \frac{9i}{n} \right) \cdot \frac{3}{n}
\]
Distribute \( \frac{3}{n} \):
\[
\sum_{i=1}^{n} \left( \frac{12}{n} + \frac{27i}{n^2} \right)
\]
Now split the sum into two parts:
\[
\sum_{i=1}^{n} \frac{12}{n} + \sum_{i=1}^{n} \frac{27i}{n^2}
\]

### Step 3: Use sum formulas
For the first part:
\[
\sum_{i=1}^{n} \frac{12}{n} = \frac{12}{n} \cdot n = 12
\]

For the second part, use the formula for the sum of the first \( n \) integers:
\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]
Thus:
\[
\sum_{i=1}^{n} \frac{27i}{n^2} = \frac{27}{n^2} \cdot \frac{n(n+1)}{2} = \frac{27(n+1)}{2n}
\]

### Step 4: Take the limit as \( n \to \infty \)
The total sum is:
\[
12 + \frac{27(n+1)}{2n}
\]
As \( n \to \infty \), \( \frac{n+1}{n} \to 1 \), so:
\[
\lim_{n \to \infty} \left( 12 + \frac{27(n+1)}{2n} \right) = 12 + \frac{27}{2} = 12 + 13.5 = 25.5
\]

Thus, the exact area under the curve is:

\[
\boxed{25.5}
\]

This is the area of the region bounded by the graph of \( f(x) = 3x - 2 \) on the interval \([2, 5]\).

Would you like more details on any step or further clarification?

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Here are 5 related questions:
1. How does the limit definition of an integral differ from a Riemann sum?
2. How would the method change if we had chosen left endpoints instead of right endpoints?
3. What if the function were non-linear, such as \( f(x) = x^2 \)? How would the steps differ?
4. How can we use the Fundamental Theorem of Calculus to compute this integral more efficiently?
5. How is this method applied in finding areas between two curves?

**Tip:** When evaluating sums involving \( i \), remember common sum formulas like \( \sum_{i=1}^n i = \frac{n(n+1)}{2} \). This can simplify calculations significantly.

Explore how to find the exact area of the region bounded by the function f(x) = 3x - 2 on [2,5] using the limit definition of a definite integral. This step-by-step solution includes key calculus concepts like Riemann sums, limits, and the Fundamental Theorem of Calculus. Suitable for students in Grades 11-12 and early college.

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