What is the formula for the trapezoidal rule?

The trapezoidal rule formula is: ∫ₐᵇ f(x)dx ≈ (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)], where Δx = (b−a)/n is the step width, a and b are the lower and upper bounds, n is the number of subintervals, and xᵢ = a + i·Δx. The endpoints are weighted by 1, and all interior points are weighted by 2.

How do you calculate using the trapezoidal rule step by step?

Step 1: Calculate the step width Δx = (b−a)/n. Step 2: List the partition points x₀, x₁, ..., xₙ where xᵢ = a + i·Δx. Step 3: Evaluate f(x) at each partition point. Step 4: Apply the formula — multiply the first and last f-values by 1, all interior f-values by 2, sum everything, then multiply by Δx/2. This gives the approximate area under the curve.

How many subintervals should I use?

More subintervals = better accuracy, but more computation. Start with 4–10 for smooth functions to get a quick estimate. Use 20–50 for high precision. The error decreases proportionally to 1/n² (quadratic convergence), so doubling n reduces the error by about a factor of 4. Very oscillatory functions may require n=50+ to capture all variations.

Trapezoidal Rule Calculator

f(x) =

Use x as variable. Supports sin, cos, tan, ln, sqrt, exp, ^, *, /, +, -

Examples

Polynomial

Trigonometric

Exponential

Integration Bounds

Lower (a)

Upper (b)

Subintervals (n)4

150

Compare with

Approximated Integral

n = 4

Δx = 0.5

4.750000

f(x) = x^2 + 1 on [0, 2]

Accuracy

Error analysis for this approximation

Reference Value

Analytical antiderivative available

4.666667

Absolute Error

|Reference − Approx|

0.083333

1.785714% relative

Error Bound

|E| ≤ (b−a)³/(12n²) · max|f″(x)|

0.083333

Step-by-Step Solution

Trapezoidal rule computation with n = 4

1Calculate Step Width

Δx = (b − a) / n = (2 − 0) / 4 = 0.5

2Find Partition Points & Evaluate f(x)

x₀ = 0 → f(x₀) = 1

x₁ = 0.5 → f(x₁) = 1.25

x₂ = 1 → f(x₂) = 2

x₃ = 1.5 → f(x₃) = 3.25

x₄ = 2 → f(x₄) = 5

3Apply Trapezoidal Rule Formula

∫f(x)dx ≈ (Δx/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_n−1) + f(x_n)]

= (0.5 / 2) × [1 + 2(1.25 + 2 + 3.25) + 5]

= 4.75

Per-Trapezoid Breakdown

Individual trapezoid areas

i	xᵢ	xᵢ₊₁	f(xᵢ)	f(xᵢ₊₁)	Area
1	0	0.5	1	1.25	0.5625
2	0.5	1	1.25	2	0.8125
3	1	1.5	2	3.25	1.3125
4	1.5	2	3.25	5	2.0625
Total					4.75

Trapezoidal Approximation

Visualizing 4 trapezoids under f(x)

f(x)

Trapezoids

What Is the Trapezoidal Rule?

A numerical method for approximating definite integrals

The trapezoidal rule is a numerical integration technique that approximates the area under a curve by dividing it into trapezoids and summing their areas. Instead of finding an exact antiderivative — which is impossible for many functions — it uses a simple geometric approach that converges to the true value as you increase the number of trapezoids.

Key Insight

The trapezoidal rule approximates the curve with straight line segments. More subintervals (larger n) means the line segments hug the curve more closely — error decreases as 1/n².

No Antiderivative

Functions like e^(−x²) or sin(x)/x have no elementary antiderivative — numerical methods are essential.

Discrete Data

When you only have measured data points rather than a formula, the trapezoidal rule works directly.

Quick Estimates

Get a fast approximation without complex symbolic computation — ideal for engineering checks.

Verification

Cross-check symbolic integration results or validate numerical simulations with a simple method.

Trapezoidal Rule Formula

The mathematical foundation and variable definitions

General Formula

Δx = (b − a) / n

∫_a^b f(x) dx ≈ (Δx/2) [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_n−1) + f(x_n)]

a, b

Integration bounds

Number of trapezoids

Δx

Step width (b−a)/n

xᵢ

Partition points

Error Bound Formula

|E| ≤ ((b−a)³ / (12n²)) · max|f″(x)| on [a,b]

Doubling n reduces the error bound by a factor of 4 (quadratic convergence). For smooth functions, even modest n gives excellent accuracy.

Worked Example

Trapezoidal rule applied to f(x) = x² + 1 on [0, 2] with n = 4

Calculate Step Width

Δx = (b − a) / n = (2 − 0) / 4 = 0.5

Δx = 0.5

List Partition Points & Evaluate f(x)

x₀ = 0 → f(0) = 0² + 1 = 1

x₁ = 0.5 → f(0.5) = 0.5² + 1 = 1.25

x₂ = 1 → f(1) = 1² + 1 = 2

x₃ = 1.5 → f(1.5) = 1.5² + 1 = 3.25

x₄ = 2 → f(2) = 2² + 1 = 5

Apply the Trapezoidal Rule Formula

∫f(x)dx ≈ (Δx/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + 2f(x₃) + f(x₄)]

= (0.5/2) × [1 + 2(1.25) + 2(2) + 2(3.25) + 5]

= 0.25 × [1 + 2.5 + 4 + 6.5 + 5]

= 0.25 × 19

= 4.75 (exact = 4.667, error = 1.79%)

Trapezoidal vs. Other Methods

How the trapezoidal rule compares to midpoint and Simpson's rules

Trapezoidal

Error: ∝ 1/n²

Uses straight lines between endpoints. Simple, works for any n. Best for periodic functions over full periods.

Midpoint

Error: ∝ 1/n²

Uses function value at the middle of each subinterval. Often more accurate than trapezoidal for the same n.

Simpson's

Error: ∝ 1/n⁴

Uses parabolic arcs. Much faster convergence but requires an even number of subintervals.

Real-World Applications

Where the trapezoidal rule is used in practice

Pharmacokinetics

AUC (Area Under the Curve) calculations for drug concentration over time in clinical trials and dosing.

Surveying

Computing land parcel areas from irregular boundary measurements using the trapezoidal formula.

Engineering

Numerical integration in FEA (Finite Element Analysis) for stress, strain, and heat transfer simulations.

Physics

Computing work done by a variable force: W = ∫F(x)dx where force changes with position.

Economics

Consumer and producer surplus calculations from supply and demand curves.

Signal Processing

Discrete-time integration of sensor data, accelerometer readings, and audio waveforms.

Common Mistakes to Avoid

Errors to watch for when using numerical integration

Using too few intervals

n = 1 gives a very rough single-trapezoid estimate. For smooth functions use at least n = 4–10; for oscillatory functions use n = 20+.

Confusing with rectangular rule

The rectangular rule uses flat tops. Trapezoidal connects endpoints with slanted lines — generally more accurate for the same n.

Forgetting function domain

Ensure your function is defined across the entire interval. ln(x) at x = 0 or sqrt(x) for x < 0 will produce errors.

Assuming Simpson's is always better

Simpson's rule requires even n and smooth functions. For periodic or noisy data, trapezoidal can be more robust.

Frequently Asked Questions

Common questions about the trapezoidal rule and numerical integration

The trapezoidal rule is a numerical integration method used to approximate definite integrals. It's particularly useful when a function has no elementary antiderivative (like e^(−x²) or sin(x)/x), when you only have discrete data points rather than a formula, or when you need a quick, reliable approximation without complex symbolic computation. It's widely used in engineering, physics, pharmacokinetics (AUC calculations), and surveying.

The error bound for the trapezoidal rule is |E| ≤ ((b−a)³/(12n²)) · max|f″(x)| on the interval [a,b]. This means: (1) Error decreases quadratically with n — double the intervals, quarter the error bound. (2) Error is proportional to the interval length cubed — wider intervals need more subintervals. (3) Error depends on the function's curvature (second derivative) — highly curved functions need more subintervals for the same accuracy.

Simpson's rule is generally more accurate for smooth functions (error ∝ 1/n⁴ vs 1/n² for trapezoidal), but the trapezoidal rule has advantages: (1) It works for any n (Simpson's requires even n). (2) It's more robust for periodic functions integrated over full periods. (3) It's simpler to implement and understand. (4) For functions with limited smoothness or noise, trapezoidal can be more reliable since Simpson's can overfit wiggles.

Yes, the trapezoidal rule works with discrete data points (x, y pairs) using the same formula: Area = sum of 0.5 × (xᵢ₊₁ − xᵢ) × (yᵢ + yᵢ₊₁). This calculator accepts function expressions; for table-based data, you can manually apply the formula with your data points. The method works for both equally and unequally spaced intervals, making it ideal for experimental data and sensor readings.

Both approximate area under a curve with n subintervals, but they differ in approach: (1) Trapezoidal uses the function values at the endpoints of each subinterval and connects them with a straight line (forming a trapezoid). (2) Midpoint uses the function value at the center of each subinterval and forms a rectangle of that height. The midpoint rule is often more accurate for the same n (error ∝ 1/n² but with a smaller constant), but the trapezoidal rule gives you endpoints that can be directly compared with data.

It depends on the function's concavity: (1) For concave-up functions (f″(x) > 0), the trapezoidal rule overestimates — the straight line between endpoints lies above the curve. (2) For concave-down functions (f″(x) < 0), it underestimates — the straight line lies below the curve. (3) For linear functions (f″(x) = 0), the trapezoidal rule gives the exact answer regardless of n. You can see this visually in the trapezoid chart above.

This calculator supports a wide range of mathematical expressions with x as the variable. You can use: polynomials (x^2 + 3*x + 1), trigonometric functions (sin(x), cos(x), tan(x)), exponential (e^x, exp(x)), logarithms (ln(x), log(x)), square roots (sqrt(x)), absolute value (abs(x)), and combinations with arithmetic operators (+, -, *, /, ^). Constants like pi and e are also supported. The calculator will attempt symbolic integration for comparison when possible.

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Last updated May 6, 2026