compute\:the\:value\:of\:the\:definite\:integral\:\int _{-1}^1\:\left(x+2\right)dx\:as\:a\:limit\:of\:a\:Reimann\:sum

To compute the value of the definite integral \(\int_{-1}^1 (x+2) \, dx\) as a limit of a Riemann sum, we can follow these steps:

### Step 1: Partition the Interval
The interval of integration is \([-1, 1]\). We partition this interval into \(n\) subintervals, each of equal width \(\Delta x\).

\[
\Delta x = \frac{1 - (-1)}{n} = \frac{2}{n}
\]

### Step 2: Choose Sample Points
For each subinterval, we need to choose a sample point \(x_i^*\). A common choice is the right endpoint, so we set:

\[
x_i^* = -1 + i\Delta x
\]

where \(i = 1, 2, \dots, n\).

### Step 3: Express the Function
The function we're integrating is \(f(x) = x + 2\). We evaluate this function at the sample points \(x_i^*\):

\[
f(x_i^*) = x_i^* + 2 = \left(-1 + \frac{2i}{n}\right) + 2 = \frac{2i}{n} + 1
\]

### Step 4: Write the Riemann Sum
The Riemann sum for this partition is:

\[
S_n = \sum_{i=1}^n f(x_i^*) \Delta x = \sum_{i=1}^n \left(\frac{2i}{n} + 1\right) \cdot \frac{2}{n}
\]

Expanding the sum:

\[
S_n = \frac{2}{n} \sum_{i=1}^n \left(\frac{2i}{n} + 1\right) = \frac{2}{n} \left(\frac{2}{n} \sum_{i=1}^n i + \sum_{i=1}^n 1\right)
\]

We know the sum formulas:

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

Substituting these into the sum:

\[
S_n = \frac{2}{n} \left(\frac{2}{n} \cdot \frac{n(n+1)}{2} + n\right) = \frac{2}{n} \left(\frac{2n(n+1)}{2n} + n\right) = \frac{2}{n} \left(n+1 + n\right) = \frac{2}{n} \cdot 2n = 4
\]

### Step 5: Take the Limit
The definite integral is the limit of the Riemann sum as \(n \to \infty\):

\[
\int_{-1}^1 (x+2) \, dx = \lim_{n \to \infty} S_n = 4
\]

So, the value of the definite integral is \(4\).

Would you like any more details or have any questions about this process? Here are some related questions you might consider:

1. How do you compute the definite integral using the Fundamental Theorem of Calculus?
2. What are the different types of Riemann sums?
3. How does the choice of sample points (left, right, midpoint) affect the Riemann sum?
4. Can you generalize this approach for any linear function \(f(x) = ax + b\)?
5. How do we interpret the geometric meaning of the definite integral?

**Tip:** Riemann sums are a fundamental concept in understanding the idea of integration. The more you practice with different functions and intervals, the more intuitive the process becomes.

Learn how to compute the definite integral \(\int_{-1}^1 (x+2) \, dx\) using the limit of a Riemann sum. Follow a detailed step-by-step solution involving partitioning the interval, choosing sample points, and evaluating the Riemann sum. Suitable for advanced high school and college students.

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