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

Question

UpStudy AI · Accepted Answer

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.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Calculus Questions