Numerical Integration — SL 5.6 Numerical integration and...

SECTION 1

Historical Context & Motivation

Long before the invention of calculus, mathematicians grappled with a deceptively simple question: How do you find the area of a shape with curved boundaries? Circles, parabolas, and irregular plots of land all posed this challenge. The earliest strategy was brilliantly practical — break the curved region into many small pieces whose areas you can calculate, then add them up. This idea is the beating heart of numerical integration, and it remains essential today whenever an exact antiderivative is impossible or impractical to find.

~250 BCE

Archimedes & Exhaustion

Archimedes estimated the area of a parabolic segment by filling it with ever-smaller triangles — an early form of numerical approximation that anticipated integral calculus by nearly two millennia.

1612

Kepler's Wine Barrels

Johannes Kepler approximated the volumes of wine barrels by slicing them into thin discs. His work in Nova Stereometria applied systematic numerical methods to practical measurement.

1687

Newton & Leibniz

The Fundamental Theorem of Calculus provided exact integration methods, but many real-world functions still had no closed-form antiderivatives, so numerical methods remained crucial.

1859

The Trapezoidal & Simpson's Rules

Formalised numerical rules — the trapezoidal rule and Simpson's rule — became standard tools in engineering and science, offering systematic accuracy improvements.

1970s–Today

Technology-Supported Integration

Graphing calculators and computer algebra systems (CAS) make numerical integration accessible to everyone, performing millions of calculations per second to approximate integrals to high precision.

In the IB Applications and Interpretation course, you are encouraged to use technology — your GDC (graphing display calculator) or software — to evaluate definite integrals numerically. The central question this topic addresses is: How can we reliably estimate the area under a curve when we don't have (or don't need) an algebraic antiderivative?

SECTION 2

Core Principles & Definitions

Before diving into calculations, it helps to nail down a few foundational ideas. A definite integral represents the net signed area between a function's curve and the x-axis over a specific interval [a, b]. When an exact antiderivative is unavailable, numerical integration steps in as a collection of techniques that approximate this area using simple geometric shapes whose areas are easy to compute.

Definite Integral

The exact net area between a curve y = f(x) and the x-axis from x = a to x = b. Written as ∫ from a to b of f(x) dx.

Numerical Approximation

An estimate of the definite integral obtained by dividing the interval into smaller sub-intervals and summing the areas of simple shapes (rectangles, trapezoids, etc.).

Sub-interval (Strip)

Each small section of the interval [a, b]. If you use n equal strips, each has width h = (b − a) / n. More strips generally means better accuracy.

Technology-Supported Evaluation

Using a GDC or software to compute ∫ from a to b of f(x) dx directly. The calculator uses its own numerical algorithm internally and returns a decimal answer.

✦ KEY TAKEAWAY

Think of numerical integration like estimating the area of an oddly-shaped swimming pool. You can't use a single length × width formula because the edges are curved. Instead, you stretch a measuring tape across the pool at regular intervals, measure the width at each point, and multiply each width by the spacing between your tapes. Add them all up and you have a solid approximation of the pool's area — the more tapes you use, the closer you get to the true answer.

SECTION 3

Visualising Numerical Integration

The diagram below shows the core idea behind numerical integration. A smooth curve y = f(x) is drawn over the interval [a, b], and the region between the curve and the x-axis is approximated using rectangles. Notice how some rectangles overshoot the curve (overestimate) and others fall short (underestimate). As you increase the number of rectangles, the total error shrinks, and the approximation improves.

Each violet rectangle has width h and height equal to f(x) at its left edge. The cyan curve is the actual function. The pink dashed lines highlight the error — the gap between the rectangle tops and the curve. Using more, narrower rectangles reduces this error.

In the diagram, the six rectangles use the left Riemann sum — the height of each rectangle is determined by the function value at the left edge of each strip. You could also use the right edge, the midpoint, or even trapezoids. Each variation gives a slightly different estimate, but they all converge to the true integral as the number of strips grows.

SECTION 4

Mathematical Framework

At the SL level, you are not required to memorise complicated formulas for numerical integration — your GDC handles the heavy lifting. However, understanding the underlying structure helps you interpret results, estimate reasonableness, and answer exam questions about approximation.

DEFINITE INTEGRAL

∫ₐᵇ f(x) dx = lim(n→∞) Σᵢ₌₁ⁿ f(xᵢ) × h

Where a and b are the limits of integration, n is the number of sub-intervals, h = (b − a) / n is the width of each strip, and xᵢ is a sample point in the i-th sub-interval.

TRAPEZOIDAL RULE (n STRIPS)

∫ₐᵇ f(x) dx ≈ (h/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + ⋯ + 2f(xₙ₋₁) + f(xₙ)]

Each pair of consecutive points defines a trapezoid. The first and last y-values count once; all middle y-values are doubled. This formula is in the IB formula booklet.

STRIP WIDTH

h = (b − a) / n

This gives the uniform width of each sub-interval. For example, if a = 1, b = 5, and n = 4, then h = (5 − 1) / 4 = 1.

🧮 GDC Tip

On a TI-84, use fnInt(f(x), x, a, b) from the MATH menu. On a TI-Nspire, go to Menu → Calculus → Numerical Integral. On a Casio fx-CG50, use the ∫ function from the RUN mode or from the graph integration feature. The calculator returns a decimal approximation — always round appropriately to the number of significant figures the question asks for.

SECTION 5

Methods of Approximation Compared

There are several geometric strategies for approximating the area under a curve. The diagram below compares three common approaches applied to the same function and interval. Understanding these differences helps you appreciate why the trapezoidal rule appears in your formula booklet, and why your GDC can give such precise answers.

Three methods applied to the same curve with 4 strips. Left rectangles underestimate for an increasing function, right rectangles overestimate, and the trapezoidal rule gives the closest approximation.

Comparison of approximation methods for SL 5.6

Method	Shape Used	Accuracy	On IB Exam?
Left/Right Riemann Sum	Rectangles	Low (large error unless many strips)	Conceptual understanding only
Trapezoidal Rule	Trapezoids	Moderate (better with more strips)	Yes — formula in booklet
GDC Numerical Integral	Algorithm (internal)	Very high (many decimal places)	Yes — expected tool

SECTION 6

Worked Example

Let's walk through a complete problem using both the trapezoidal rule and a GDC to approximate the integral of f(x) = x² + 1 from x = 0 to x = 4 using 4 strips.

Trapezoidal Rule: ∫₀⁴ (x² + 1) dx with n = 4

Step 1 — Calculate the strip width hUsing h = (b − a) / n = (4 − 0) / 4 = 1. Each strip is 1 unit wide.

h = 1

Step 2 — Identify the x-valuesStarting at x₀ = 0 and adding h each time: x₀ = 0, x₁ = 1, x₂ = 2, x₃ = 3, x₄ = 4. That gives us n + 1 = 5 x-values.

x = {0, 1, 2, 3, 4}

Step 3 — Evaluate f(x) at each x-valuef(0) = 0² + 1 = 1, f(1) = 1² + 1 = 2, f(2) = 2² + 1 = 5, f(3) = 3² + 1 = 10, f(4) = 4² + 1 = 17.

y-values: {1, 2, 5, 10, 17}

Step 4 — Apply the trapezoidal rule formula∫₀⁴ f(x) dx ≈ (h/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + 2f(x₃) + f(x₄)] = (1/2) × [1 + 2(2) + 2(5) + 2(10) + 17] = (1/2) × [1 + 4 + 10 + 20 + 17] = (1/2) × 52 = 26.

Trapezoidal approximation = 26

Step 5 — Compare with the GDC / exact answerThe exact value is ∫₀⁴ (x² + 1) dx = [x³/3 + x]₀⁴ = (64/3 + 4) − 0 = 76/3 ≈ 25.333. Our trapezoidal estimate of 26 is an overestimate by about 0.667. Using your GDC, you would enter the function and limits to obtain 25.333… directly. The trapezoidal rule gave a reasonable approximation with only 4 strips, but the GDC is more precise.

Exact value ≈ 25.333; GDC confirms this

💡 Why was the estimate too high?

The function x² + 1 is concave up (curves upward) on [0, 4]. For concave-up functions, the straight-line tops of the trapezoids sit above the curve, so the trapezoidal rule overestimates the area. Increasing n from 4 to, say, 20 strips would significantly reduce this error.

SECTION 7

Strengths & Limitations

Numerical integration is incredibly versatile, but it's important to understand when it shines and where its weaknesses lie. Knowing these trade-offs will help you choose the right approach on exams and in real-world applications.

Trade-offs of numerical integration

Strengths	Limitations
Works for ANY continuous function — you don't need to find an antiderivative	Produces an approximation, not an exact answer (though it can be extremely close)
Handles data given in table form (e.g., experimental measurements) where no formula exists	Accuracy depends on the number of strips — too few strips means a large error
GDC technology makes it fast, accessible, and highly accurate	Functions with sharp corners, discontinuities, or rapid oscillation can cause larger errors
Can be applied in contexts like area, distance, accumulated change, and probability	For the trapezoidal rule by hand, calculations can become tedious with many strips

✦ KEY TAKEAWAY

Think of the trapezoidal rule as a GPS route estimate and the GDC as turning on live traffic data. The GPS gives you a decent estimate using straight-line segments between known points. Live traffic (more data, finer resolution) refines the estimate. Both approaches get you where you're going — it's just a matter of how precise you need to be.

SECTION 8

Connection to Advanced Theory

At the SL level you focus on using technology and the trapezoidal rule to approximate integrals. But these ideas are the gateway to much deeper territory in HL and university-level mathematics. Here's a quick look at how SL concepts connect to more advanced ideas.

How SL topics extend into HL and beyond

SL 5.6 (This Course)	HL / University Extension
Trapezoidal rule with given n	Simpson's rule (uses parabolic arcs instead of lines for better accuracy)
GDC numerical integral	Gaussian quadrature and adaptive algorithms that automatically choose strip widths
Recognising that more strips = better accuracy	Error bounds — quantifying the maximum possible error for a given n
Approximating area under a curve	Monte Carlo integration — using random sampling for high-dimensional integrals in physics and finance

For now, your focus should be on confidently setting up the trapezoidal rule, interpreting your GDC output, and understanding what a numerical integral means geometrically — the signed area between a curve and the x-axis. These skills form the foundation for everything that follows.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

Explain in your own words why numerical integration is necessary. Give one example of a situation where you would need it instead of finding an exact antiderivative.

PROBLEM 2 — BASIC CALCULATION

Use the trapezoidal rule with n = 3 strips to approximate ∫₁⁴ (2x + 1) dx. Then find the exact value and compare.

PROBLEM 3 — INTERMEDIATE

The table below gives values of a function g(x). Use the trapezoidal rule to estimate ∫₀⁸ g(x) dx. | x | 0 | 2 | 4 | 6 | 8 | | g(x) | 3 | 7 | 8 | 6 | 4 |

PROBLEM 4 — APPLIED

A river's cross-section is measured at 5 equally-spaced points across its width of 8 metres. The depths (in metres) at x = 0, 2, 4, 6, 8 are: 0, 1.8, 3.2, 2.4, 0. Use the trapezoidal rule to estimate the cross-sectional area of the river, and explain what this represents physically.

PROBLEM 5 — CRITICAL THINKING

A student uses the trapezoidal rule with 4 strips to approximate ∫₀² f(x) dx for a concave-down function and gets 12.5. They then repeat with 8 strips and get 12.3. (a) Is the trapezoidal rule overestimating or underestimating for a concave-down function? (b) Which estimate is closer to the true value, and why? (c) What would happen if they used 100 strips?

SUMMARY

Lesson Summary

Numerical integration approximates the definite integral — the net area between a curve and the x-axis over an interval [a, b]. The trapezoidal rule divides the interval into n equal strips of width h = (b − a) / n and approximates the area using trapezoids: (h/2) × [first y + 2 × (middle y-values) + last y]. More strips mean smaller errors. For concave-up functions the rule overestimates; for concave-down functions it underestimates.

In the IB Applications and Interpretation SL course, you are expected to use your GDC to evaluate definite integrals numerically — it applies sophisticated internal algorithms to return a highly accurate decimal result. The trapezoidal rule is the manual method you should be able to apply by hand, especially when given tabular data (experimental measurements) rather than a formula. Together, these tools let you find areas, accumulated quantities, and total change for virtually any function you'll encounter.