Lesson 4: Numerical Integration
Estimated time: 35-45 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Apply the Trapezoidal Rule to approximate definite integrals
- Apply Simpson's Rule for higher-accuracy approximation
- Understand error bounds for each method
- Compare the methods and know when to use each
Why Numerical Integration?
Many important integrals cannot be expressed in terms of elementary functions. Examples include ∫ e−x² dx (the Gaussian integral), ∫ sin(x²) dx, and ∫ √(1 + x4) dx. For these, we need numerical methods to approximate the value.
Even when an antiderivative exists, numerical methods can be faster for one-time computations or when the integrand is given as a table of data points.
The Trapezoidal Rule
Instead of approximating the area with rectangles (Riemann sums), the Trapezoidal Rule uses trapezoids, which fit the curve more closely.
Trapezoidal Rule
With n subintervals and Δx = (b − a)/n:
Tn = (Δx/2)[f(x0) + 2f(x1) + 2f(x2) + ... + 2f(xn−1) + f(xn)]
Note the pattern: first and last values get weight 1; all interior values get weight 2.
Example 1: Trapezoidal Rule
Approximate ∫02 e−x² dx with n = 4.
Step 1: Δx = 2/4 = 0.5. Points: x0=0, x1=0.5, x2=1, x3=1.5, x4=2.
Step 2: Evaluate: f(0) = 1, f(0.5) = 0.7788, f(1) = 0.3679, f(1.5) = 0.1054, f(2) = 0.0183.
Step 3: T4 = (0.5/2)[1 + 2(0.7788) + 2(0.3679) + 2(0.1054) + 0.0183]
= 0.25[1 + 1.5576 + 0.7358 + 0.2108 + 0.0183] = 0.25(3.5225) ≈ 0.8806.
Simpson's Rule
Simpson's Rule approximates the integrand with parabolas instead of straight lines, yielding much better accuracy. It requires n to be even.
Simpson's Rule
With n subintervals (n even) and Δx = (b − a)/n:
Sn = (Δx/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + ... + 4f(xn−1) + f(xn)]
Pattern: 1, 4, 2, 4, 2, ..., 4, 1.
Example 2: Simpson's Rule
Approximate ∫02 e−x² dx with n = 4 (same as Example 1).
S4 = (0.5/3)[1 + 4(0.7788) + 2(0.3679) + 4(0.1054) + 0.0183]
= (1/6)[1 + 3.1152 + 0.7358 + 0.4216 + 0.0183] = (1/6)(5.2909) ≈ 0.8818.
(The true value is approximately 0.8821. Simpson's is already very close with just n = 4!)
Error Bounds
Error Bound for Trapezoidal Rule
|ET| ≤ M(b − a)³ / (12n²), where M = max |f''(x)| on [a, b].
Error decreases as O(1/n²): doubling n cuts the error by 4.
Error Bound for Simpson's Rule
|ES| ≤ K(b − a)5 / (180n4), where K = max |f(4)(x)| on [a, b].
Error decreases as O(1/n4): doubling n cuts the error by 16!
Example 3: Comparing Accuracy
For ∫01 x³ dx (exact answer = 1/4), compare T4 and S4.
Δx = 0.25. Points: 0, 0.25, 0.5, 0.75, 1. Values: 0, 0.0156, 0.125, 0.4219, 1.
T4 = (0.25/2)[0 + 2(0.0156) + 2(0.125) + 2(0.4219) + 1] = 0.125(2.1250) = 0.2656. Error: 0.0156.
S4 = (0.25/3)[0 + 4(0.0156) + 2(0.125) + 4(0.4219) + 1] = (1/12)(3.0000) = 0.2500. Error: 0!
Simpson's Rule gives the exact answer for cubics (because f(4) = 0 for polynomials of degree 3 or less).
Working with Data Tables
Numerical methods are essential when you only have data points (no formula).
Example 4: From a Data Table
Estimate ∫06 f(x) dx using Simpson's Rule given:
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| f(x) | 3 | 5 | 4 | 6 | 7 | 5 | 2 |
n = 6 (even), Δx = 1.
S6 = (1/3)[3 + 4(5) + 2(4) + 4(6) + 2(7) + 4(5) + 2] = (1/3)[3 + 20 + 8 + 24 + 14 + 20 + 2] = 91/3 ≈ 30.33.
Comparison and Guidelines
| Method | Accuracy | Requirement | Best For |
|---|---|---|---|
| Trapezoidal | O(1/n²) | Any n | Quick estimates, odd n |
| Simpson's | O(1/n4) | n must be even | Best accuracy per function evaluation |
Rule of thumb: Simpson's Rule with n = 10 often matches the Trapezoidal Rule with n = 100 in terms of accuracy.
Key Takeaways
- Trapezoidal Rule: weighted average with pattern 1, 2, 2, ..., 2, 1. Error is O(1/n²).
- Simpson's Rule: pattern 1, 4, 2, 4, 2, ..., 4, 1. Requires even n. Error is O(1/n4).
- Simpson's is exact for polynomials of degree 3 or less.
- Both methods work with data tables when no formula is available.
Check Your Understanding
1. Use the Trapezoidal Rule with n = 4 to approximate ∫13 (1/x) dx.
2. Use Simpson's Rule with n = 4 for the same integral.
3. If you need the Trapezoidal Rule error to be less than 0.001 for ∫01 ex dx, how many subintervals do you need?