To find the area of the region bounded by $f(x) = 3x - 2$ and the x-axis over the interval $[2,5]$ using the limit definition, we can proceed as follows:

Step 1: Set up the area formula using the limit definition

The area under a curve can be found using the limit of a Riemann sum. The interval $[a, b]$ is divided into $n$ subintervals, each with width $\Delta x$, defined as:

$\Delta x = \frac{b - a}{n}$

For this problem:

• $a = 2$
• $b = 5$
• Therefore, $\Delta x = \frac{5 - 2}{n} = \frac{3}{n}$

Next, we define the $i$-th subinterval and the corresponding x-values. The $i$-th x-value is given by:

$x_i = a + i \Delta x = 2 + i \cdot \frac{3}{n}$

Step 2: Define the Riemann sum

The Riemann sum approximating the area is given by:

$\sum_{i=1}^{n} f(x_i) \cdot \Delta x$

Plugging in the function $f(x) = 3x - 2$:

$\sum_{i=1}^{n} \left[ 3 \left( 2 + i \cdot \frac{3}{n} \right) - 2 \right] \cdot \Delta x$

Simplifying:

[ \sum_{i=1}^{n} \left[ 6 + \frac{9i}{n} - 2 \right] \cdot \Delta x = \sum_{i=1}^{n} \left[ 4 + \frac{9i}{n} \right] \cdot \frac{3}{n} ]

$= \sum_{i=1}^{n} \left[ \frac{12}{n} + \frac{27i}{n^2} \right]$

Step 3: Calculate the limit

We now take the limit as $n \to \infty$:

$\lim_{n \to \infty} \sum_{i=1}^{n} \left[ \frac{12}{n} + \frac{27i}{n^2} \right]$

Separate the sum:

$\lim_{n \to \infty} \left[ \sum_{i=1}^{n} \frac{12}{n} + \sum_{i=1}^{n} \frac{27i}{n^2} \right]$

Calculate each sum:

1. Sum of a constant: $\sum_{i=1}^{n} \frac{12}{n} = 12$

2. Sum of $i$: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2} \approx \frac{n^2}{2}$ for large $n$.

Thus:

[ \lim_{n \to \infty} \left[ 12 + \frac{27 \cdot n^2}{2 }\