The Midpoint Riemann Sum is a method for approximating the definite integral of a function. It's particularly useful when you don't have an easy antiderivative or when dealing with complex functions. This guide will walk you through calculating a Midpoint Riemann Sum using a table, making the process clear and straightforward.
Understanding the Midpoint Riemann Sum
The core idea behind the Midpoint Riemann Sum is to divide the area under a curve into several rectangles. Unlike the left or right Riemann sums, the height of each rectangle is determined by the function's value at the midpoint of each subinterval. This often leads to a more accurate approximation, especially for smoothly varying functions.
Key Components:
- Function: The function you're integrating (e.g., f(x) = x²).
- Interval: The range of x-values over which you're integrating (e.g., [a, b]).
- Number of Subintervals (n): The number of rectangles you'll use to approximate the area. More subintervals generally lead to a more accurate approximation, but also require more calculations.
- Width of each Subinterval (Δx): This is calculated as (b - a) / n. It represents the width of each rectangle.
- Midpoints: The x-values at the center of each subinterval.
Step-by-Step Calculation with a Table
Let's illustrate with an example. We'll approximate the integral of f(x) = x² from x = 1 to x = 5 using 4 subintervals (n = 4).
1. Calculate Δx:
Δx = (5 - 1) / 4 = 1
2. Determine the Subintervals:
With Δx = 1, our subintervals are [1, 2], [2, 3], [3, 4], and [4, 5].
3. Find the Midpoints:
The midpoints of each subinterval are: 1.5, 2.5, 3.5, and 4.5.
4. Create a Table:
Now, let's organize our calculations in a table:
| Subinterval | Midpoint (xᵢ) | f(xᵢ) = xᵢ² | Area of Rectangle (f(xᵢ) * Δx) |
|---|---|---|---|
| [1, 2] | 1.5 | 2.25 | 2.25 |
| [2, 3] | 2.5 | 6.25 | 6.25 |
| [3, 4] | 3.5 | 12.25 | 12.25 |
| [4, 5] | 4.5 | 20.25 | 20.25 |
| Total | 41 |
5. Calculate the Midpoint Riemann Sum:
The Midpoint Riemann Sum is the sum of the areas of all the rectangles. In our table, this is simply the sum of the last column: 2.25 + 6.25 + 12.25 + 20.25 = 41.
Therefore, the approximation of the integral of x² from 1 to 5 using the Midpoint Riemann Sum with 4 subintervals is 41.
Increasing Accuracy
Remember, increasing the number of subintervals (n) generally improves the accuracy of the approximation. Try repeating the process with a larger 'n' to see how the approximation converges towards the true value of the integral (which, in this case, is 124/3 ≈ 41.33).
Applying to More Complex Functions
This table method works equally well for more complex functions. Simply replace the f(xᵢ) column with the calculations for your specific function at each midpoint.
Conclusion
The Midpoint Riemann Sum, coupled with a well-organized table, provides a clear and efficient way to approximate definite integrals. By understanding the underlying principles and following these steps, you can accurately estimate the area under a curve, even for complex functions, using this powerful numerical integration technique. Remember that using more subintervals typically yields a more precise approximation.
