Numerical Integration: Approximating Definite Integrals

Learning Objectives

By the end of this study guide, you should be able to:

• Approximate the value of a definite integral using the midpoint and trapezoidal rules.
• Determine the absolute and relative error when utilizing numerical integration techniques.
• Estimate the absolute and relative error using standardized error-bound formulas.
• Recognize graphical conditions that cause the midpoint and trapezoidal rules to over- or underestimate the true value.
• Use Simpson’s rule to approximate definite integrals to a higher degree of given accuracy.

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Key Terms & Glossary

• Numerical Integration: A family of computational methods used to estimate the value of a definite integral when exact closed-form antiderivatives are impossible or impractical to find.
• Absolute Error: The precise magnitude of the difference between an estimated value and the exact actual value.
• Relative Error: The absolute error expressed as a fraction or percentage of the exact actual value, providing context to the error's scale.
• Midpoint Rule: A numerical approximation method that estimates the area under a curve using Riemann sum rectangles, where the height of each rectangle is evaluated at the exact midpoint of its subinterval.
• Trapezoidal Rule: An approximation rule that calculates the area under a curve by dividing the region into vertical trapezoids rather than flat-topped rectangles.
• Simpson’s Rule: An advanced numerical method that approximates the integrand using piecewise quadratic functions (parabolas), yielding extremely accurate area estimates.

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The "Big Idea"

While the Fundamental Theorem of Calculus allows us to evaluate definite integrals easily if we can find an antiderivative, many real-world functions (like the standard normal distribution $f(x) = e^{-x^2}$) cannot be expressed in a simple closed form.

Numerical integration solves this by returning to the fundamental concept of an integral: area under a curve. By dividing the area into familiar geometric shapes (rectangles, trapezoids, or parabolas), we can calculate a highly accurate numerical estimate. The more slices (subintervals) we use, the smaller our approximation error becomes.

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Formula / Concept Box

[!TIP] Pattern Memorization Notice the specific coefficient patterns in the brackets:

• Trapezoidal: 1, 2, 2, 2... 2, 1
• Simpson's: 1, 4, 2, 4, 2... 4, 1

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Hierarchical Outline

• 1. The Need for Numerical Integration
  • Limitations of closed-form antiderivatives (e.g., complex integrals).
  • Use cases in real-world data (integrating discrete experimental data points).
• 2. Linear Approximations
  • The Midpoint Rule: Evaluating Riemann sums at interval centers.
  • The Trapezoidal Rule: Averaging left and right Riemann sums using trapezoids.
  • Error Behavior: Concavity determines over/underestimation.
• 3. Quadratic Approximation
  • Simpson's Rule: Connecting points via piecewise parabolas.
  • Requirement: Number of subintervals ($n$) must be even.
• 4. Error Analysis
  • Calculating Absolute Error.
  • Calculating Relative Error.
  • Utilizing error-bound formulas to guarantee approximation precision.

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Visual Anchors

1. Strategy Decision Flow

2. Geometric Representation: Trapezoidal Rule

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Definition-Example Pairs

• Midpoint Rule
  • Definition: Estimating integral area using rectangles whose heights touch the curve precisely at the horizontal middle of the interval.
  • Real-World Example: Estimating total water consumption over a day by checking the flow rate exactly at the top/middle of every hour (12:30, 1:30, etc.) rather than the start or end.
• Trapezoidal Rule
  • Definition: Estimating area by connecting the function's endpoints on each subinterval with a straight line, forming a trapezoid.
  • Real-World Example: Calculating the total distance a car traveled using speedometer readings taken every 10 seconds, assuming acceleration was steady (linear) between those 10-second checks.
• Absolute vs. Relative Error
  • Definition: Absolute error is the raw numerical discrepancy; Relative error scales that discrepancy against the true value.
  • Real-World Example: If an engineer estimates a bridge cable needs to be 100 meters, but it actually needs to be 101 meters, the absolute error is 1 meter. The relative error is $1/101 \approx 0.99%$. A 1-meter absolute error on a 5-meter component, however, is a massive 20% relative error!

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Comparison Tables

[!WARNING] Don't Forget! Simpson's Rule cannot be used if you have an odd number of subintervals ($n$). The formula relies on pairing intervals to build quadratic fits.

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Worked Examples

▶Example 1: Approximating with the Trapezoidal Rule (Click to Expand)

Problem: Estimate $\int_{0}^{2} x^2 \,dx$ using the Trapezoidal Rule with $n = 2 subintervals.

Step 1: Find the subinterval width (\Delta x$). $\Delta x = \frac{b - a}{n} = \frac{2 - 0}{2} = 1$

**Step 2: Identify the . , , .

Step 3: Apply the Trapezoidal Rule Formula. $T_2 = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + f(x_2)]$ $T_2 = \frac{1}{2} [ (0)^2 + 2(1)^2 + (2)^2 ]$ $T_2 = \frac{1}{2} [ 0 + 2(1) + 4 ] = \frac{1}{2} [6] = 3$

Answer: The estimated area is 3.

▶Example 2: Calculating Absolute and Relative Error (Click to Expand)

Problem: Using the result from Example 1, determine the absolute and relative error of the approximation.

Step 1: Find the exact actual value. $\int_{0}^{2} x^2 \,dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} \approx 2.6667$

Step 2: Calculate Absolute Error ($|E|$). $|E| = |\text{Actual} - \text{Estimate}|$ $|E| = |2.6667 - 3| = 0.3333$

Step 3: Calculate Relative Error. $\text{Relative Error} = \frac{0.3333}{2.6667} \approx 0.125 \text{ or } 12.5\%$

Answer: The absolute error is $0.3333, and the relative error is 12.5%.

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Checkpoint Questions

Test your knowledge with these active recall questions. If you struggle, review the sections above!

1. Why do we need numerical integration if we already have the Fundamental Theorem of Calculus?
2. In Simpson's Rule, what geometric shape is used to connect the evaluated points on the curve?
3. If the function $f(x) is entirely concave up on an interval, will the Trapezoidal Rule overestimate or underestimate the exact integral? Why?
4. What is the required condition for the number of subintervals (n) when applying Simpson’s Rule?
5. Write out the multiplier pattern used in the brackets for a Trapezoidal Rule estimation with n=4$.