Trapezoidal Rule is used for approximating definite integral.
This method works by approximating the area under the function (in given interval) as trapezoid(s).
As can be seen, the bigger the n (more intervals) is, the better the integral is approximated.
Formula:
Where:
- a is the beggining of the interval, b is the end of the interval,
- n is the number of steps (number of trapezoids to use),
- f(x) is the function to integrate,
- m is the height of i-th trapezoid.
Example:
Let’s approximate this definite integral with n = 5.
As can be seen, for n = 5, we calculated that the deinite integral is equal to 68/25 = 2.72, which is fairly close to the actual value: 8/3 = 2.(6).
Trapezoidal Rule N Comparison
| n | I | Error |
|---|---|---|
| 5 | 68/25 = 2.72 | 4/75 = 0.05(3) |
| 10 | 67/25 = 2.68 | 1/75 = 0.01(3) |
| 25 | 1668/625 = 2.6688 | 4/1875= 0.0021(3) |
Algorithm:
IN: Function f, which is continous function, interval [a,b] and number of iterations to perform - n. OUT: Defined Integral approximation of given function in given interval. 1. Calculate step = (b - a) / n. It will be used as the step size of each iteration. 2. Assign 0.5 * (f(a) + f(b)) to variable integral. 3. Initialize variable i to 0. 4. Add f(a + step * i) to variable integral. 5. Increase i by one. 6. Check if i is smaller than n. If it is go to step 4. 7. Multiply integral by step. 8. End algorithm and return integral.
Sample Output:
f(x) = x * x -2 2 1000000 integral: 5.33334933334409
Pros:
- Easy to implement
Cons:
- Requires great number of iterations to be accurate, especially for big intervals (huge value of b – a)
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Trapezoidal Rule Algorithm