Find the approximation for the area under the function between $$ x=2 $$ and $$ x=26 $$ for the given data. $$ \begin{array}{c|cc|} x \mid 246101214161820222426 \ f(x) \mid 371287910151720181611 \ M P S_{3}=\text { ? } \end{array} $$

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To find the approximation for the area under the function $$ f(x) $$ between $$ x=2 $$ and $$ x=26 $$ using the Midpoint Rule (MPR), we will follow these steps:

1. **Identify the intervals**: The given $$ x $$ values are $$ 2, 6, 10, 14, 18, 22, 26 $$. The intervals are:
   - $$ [2, 6] $$
   - $$ [6, 10] $$
   - $$ [10, 14] $$
   - $$ [14, 18] $$
   - $$ [18, 22] $$
   - $$ [22, 26] $$

2. **Calculate the width of each interval**: The width $$ \Delta x $$ for each interval is $$ 4 $$ (since $$ 6 - 2 = 4 $$, $$ 10 - 6 = 4 $$, etc.).

3. **Find the midpoints**: The midpoints for each interval are:
   - Midpoint of $$ [2, 6] $$: $$ 4 $$
   - Midpoint of $$ [6, 10] $$: $$ 8 $$
   - Midpoint of $$ [10, 14] $$: $$ 12 $$
   - Midpoint of $$ [14, 18] $$: $$ 16 $$
   - Midpoint of $$ [18, 22] $$: $$ 20 $$
   - Midpoint of $$ [22, 26] $$: $$ 24 $$

4. **Evaluate $$ f(x) $$ at the midpoints**:
   - $$ f(4) = 12 $$
   - $$ f(8) = 15 $$
   - $$ f(12) = 17 $$
   - $$ f(16) = 20 $$
   - $$ f(20) = 18 $$
   - $$ f(24) = 16 $$

5. **Calculate the area using the Midpoint Rule**:
   $$
   \text{Area} \approx \Delta x \cdot (f(4) + f(8) + f(12) + f(16) + f(20) + f(24))
   $$
   $$
   \text{Area} \approx 4 \cdot (12 + 15 + 17 + 20 + 18 + 16)
   $$

Now, let's calculate the sum and then the area.
Calculate the value by following steps:
- step0: Calculate:
  $$4\left(12+15+17+20+18+16\right)$$
- step1: Add the numbers:
  $$4\times 98$$
- step2: Multiply the numbers:
  $$392$$
The total area under the function $$ f(x) $$ between $$ x=2 $$ and $$ x=26 $$ using the Midpoint Rule is approximately $$ 392 $$.
The area under the function $$ f(x) $$ between $$ x=2 $$ and $$ x=26 $$ using the Midpoint Rule is approximately 392.

Find the approximation for the area under the function between and for the given data.

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