Approximating Definite Integrals
When an antiderivative doesn't exist (like ), we can't use the Fundamental Theorem. Instead, we approximate the area numerically.
Why Approximate?
Some integrands simply cannot be integrated using any known technique. Famous examples include:
Gaussian integral
Sine integral
Elliptic integral
Solution: Approximate the area by dividing it into simple shapes (rectangles, trapezoids, or parabolas) and adding up their areas!
Three Methods
Midpoint Rule
Use rectangles with height at the center of each subinterval.
where is the midpoint of the th subinterval
Trapezoid Rule
Connect function values with straight lines to form trapezoids.
First and last terms have coefficient 1, all middle terms have coefficient 2
Simpson's Rule
Most accurate! Requires n to be even.
Connect function values with parabolas (quadratic curves).
Pattern: 1, 4, 2, 4, 2, 4, ..., 2, 4, 1
Fun fact: Simpson's Rule is exact for polynomials up to degree 3!
Formulas at a Glance
Given with subintervals:
| Method | Coefficients | Multiplier | Accuracy |
|---|---|---|---|
| Midpoint | All 1s (at midpoints) | Good | |
| Trapezoid | 1, 2, 2, ..., 2, 1 | Better | |
| Simpson's | 1, 4, 2, 4, ..., 4, 1 | Best |
Worked Examples
Example 1: Trapezoid Rule
Approximate with using the Trapezoid Rule.
The exact value is . Error is about 0.67.
Example 2: Simpson's Rule
Approximate with using Simpson's Rule.
Same function values as before.
This is exactly ! Simpson's Rule is exact for quadratics!
Example 3: Error Bound
How many intervals are needed to approximate within 0.001 using Trapezoid Rule?
where on
, so
Maximum on is at :
We need at least 13 subintervals to guarantee error less than 0.001.
Practice Quiz
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