To apply the Mid Ordinate Rule to find the approximate value of $e^{x^2}$ using 5 ordinates, we proceed step by step:

1. Mid Ordinate Rule Formula

The Mid Ordinate Rule is used to approximate the area under a curve. The formula is:

$\text{Area} \approx h \sum_{i=1}^{n} f(x_i)$

Where:

• $h$ is the width of each interval (the distance between consecutive ordinates),
• $x_i$ are the midpoints of each interval, and
• $f(x_i)$ is the function value at each midpoint.

2. Define the Function

In this case, we are approximating the integral of $e^{x^2}$.

$f(x) = e^{x^2}$

3. Choosing the Interval and Midpoints

Assume the interval $[a, b]$ over which we are calculating the integral is known. Since you mentioned using 5 ordinates, we will divide the interval into 5 subintervals and find the midpoints.

1. Divide the range $[a, b]$ into 5 intervals, so each interval will have a width $h$.
2. The midpoints $x_1, x_2, x_3, x_4, x_5$ will be at the center of these intervals.

4. Compute the Function Values

For each midpoint $x_i$, compute $f(x_i) = e^{x_i^2}$.

5. Apply the Mid Ordinate Rule

Once you have the midpoints and the function values, use the Mid Ordinate Rule to sum the function values and multiply by the interval width $h$.

Without specific values for the interval or the function being integrated, I can't provide a numerical answer, but I can assist further if you provide an interval or clarification.

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Would you like to specify the interval, or should I use a standard one like $[0, 1]$ for demonstration? Let me know if you need more details!

Relative Questions:

1. What interval would you like to use for $e^{x^2}$?
2. Would you like to see a demonstration with specific values?
3. Are you familiar with other approximation methods like the Trapezoidal Rule?
4. How do you interpret the number of ordinates in this context?
5. Would you prefer I calculate this approximation numerically?

Tip:

When using numerical integration methods, dividing the interval into smaller sections generally improves accuracy.