Use the limit definition to find the area of the region bounded by the graph of the following function and the x-axis over the given interval. Don’t forget to sketch the region. Give an exact answer. 𝑓(𝑥) = 3𝑥 − 2 on [2,5]

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 }\

Use the limit definition to find the area of the region bounded by the graph of the following function and the x-axis over the given interval. Don’t forget to sketch the region. Give an exact answer. f(x) = 3x − 2 on [2,5]

Learn how to find the exact area of the region bounded by the linear function f(x) = 3x - 2 and the x-axis over the interval [2,5] using the limit definition of a Riemann sum. This step-by-step solution is ideal for advanced calculus students in Grade 12 or higher.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation