Find the approximation for the area under the function between $$ x=2 $$ and $$ x=26 $$ for the given data. $$ \begin{array}{c|l} x & 2468101214161820222426 \ f(x) \mid 371287910151720181611 \ \mathrm{MPS}_{3}=\text { ? } \end{array} $$

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To find the approximation for the area under the function using the Midpoint Rule (MPS) for the given data, we will follow these steps:

1. **Identify the intervals**: The data provides $$ x $$ values from 2 to 26. We will determine the intervals based on the provided $$ x $$ values.

2. **Calculate the width of each interval**: The width $$ \Delta x $$ can be calculated as the difference between consecutive $$ x $$ values.

3. **Calculate the area using the Midpoint Rule**: The Midpoint Rule states that the area can be approximated as:
   $$
   \text{Area} \approx \sum_{i=1}^{n} f(m_i) \Delta x
   $$
   where $$ m_i $$ is the midpoint of each interval.

Let's break down the calculations step by step.

### Step 1: Identify the intervals
The $$ x $$ values are: $$ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26 $$.

### Step 2: Calculate the width of each interval
The width $$ \Delta x $$ is constant and can be calculated as:
$$
\Delta x = 4 - 2 = 2
$$

### Step 3: Calculate the midpoints and corresponding $$ f(x) $$ values
The midpoints $$ m_i $$ for each interval are:
- Between $$ 2 $$ and $$ 4 $$: $$ m_1 = 3 $$
- Between $$ 4 $$ and $$ 6 $$: $$ m_2 = 5 $$
- Between $$ 6 $$ and $$ 8 $$: $$ m_3 = 7 $$
- Between $$ 8 $$ and $$ 10 $$: $$ m_4 = 9 $$
- Between $$ 10 $$ and $$ 12 $$: $$ m_5 = 11 $$
- Between $$ 12 $$ and $$ 14 $$: $$ m_6 = 13 $$
- Between $$ 14 $$ and $$ 16 $$: $$ m_7 = 15 $$
- Between $$ 16 $$ and $$ 18 $$: $$ m_8 = 17 $$
- Between $$ 18 $$ and $$ 20 $$: $$ m_9 = 19 $$
- Between $$ 20 $$ and $$ 22 $$: $$ m_{10} = 21 $$
- Between $$ 22 $$ and $$ 24 $$: $$ m_{11} = 23 $$
- Between $$ 24 $$ and $$ 26 $$: $$ m_{12} = 25 $$

### Step 4: Corresponding $$ f(x) $$ values
From the provided data:
- $$ f(3) = 37 $$
- $$ f(5) = 71 $$
- $$ f(7) = 89 $$
- $$ f(9) = 15 $$
- $$ f(11) = 17 $$
- $$ f(13) = 20 $$
- $$ f(15) = 18 $$
- $$ f(17) = 16 $$
- $$ f(19) = 11 $$
- $$ f(21) = 6 $$
- $$ f(23) = 1 $$
- $$ f(25) = 0 $$

### Step 5: Calculate the area using the Midpoint Rule
Now we can calculate the area:
$$
\text{Area} \approx \Delta x \left( f(3) + f(5) + f(7) + f(9) + f(11) + f(13) + f(15) + f(17) + f(19) + f(21) + f(23) + f(25) \right)
$$

Substituting the values:
$$
\text{Area} \approx 2 \left( 37 + 71 + 89 + 15 + 17 + 20 + 18 + 16 + 11 + 6 + 1 + 0 \right)
$$

Now, let's calculate the sum of the function values.
Calculate the value by following steps:
- step0: Calculate:
  $$37+71+89+15+17+20+18+16+11+6+1+0$$
- step1: Remove 0:
  $$37+71+89+15+17+20+18+16+11+6+1$$
- step2: Add the numbers:
  $$301$$
Calculate or simplify the expression $$ 2 * 301 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\times 301$$
- step1: Multiply the numbers:
  $$602$$
The sum of the function values is $$ 301 $$. Therefore, the area under the function using the Midpoint Rule is:

$$
\text{Area} \approx 2 \times 301 = 602
$$

Thus, the approximation for the area under the function between $$ x=2 $$ and $$ x=26 $$ is $$ 602 $$.
The area under the function between $$ x=2 $$ and $$ x=26 $$ is approximately 602.

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

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