Trapezoidal Rule Calculator

The Trapezoidal Rule Calculator estimates arc length numerically. It applies the composite trapezoidal rule to the arc-length integral of a function you enter. Follow these steps to get a first estimate in under a minute.

1. Enter the function. Type f(x) in the function field, for example x^2, sin(x), or ln(x+1). Use ^ for powers and standard function names such as sin, cos, exp, and ln.
2. Enter the starting value. Set a, the lower integration bound. For the curve segment from x = 0 to x = 1, enter 0.
3. Enter the ending value. Set b, the upper integration bound. The calculator requires b > a.
4. Choose subdivisions. Enter n, the number of equal subintervals. A first run with n = 10 works well for smooth curves.
5. Select the trapezoidal method. In the numerical integration interface, choose the trapezoidal rule so the panel formula below is the one being applied.
6. Click Calculate. The tool evaluates the arc-length integrand at n + 1 sample points, applies the endpoint weights, and displays the approximation.
7. Interpret the result. The displayed number is an estimate of the curve length between x = a and x = b, in the same length units as your x values.
8. Improve accuracy. Double n and recalculate. When two successive estimates barely change, the result has converged for practical purposes.

Here is what each input controls:

Entering f(x) = x², a = 0, b = 1, and n = 10 produces an arc-length estimate near 1.479, which is close to the exact value 1.478943. The rest of this guide explains exactly what the calculator computes and how to judge the quality of its answer.

The trapezoidal rule is a numerical integration method. It approximates a definite integral by replacing the area under a curve with the area of trapezoids drawn under the integrand. Each trapezoid is built from two adjacent sample points, and the sum of all trapezoid areas estimates the integral.

The name comes from the geometry. Between two sample points x_i and x_i+1, the integrand g(x) is replaced by a straight line. The region under that straight line is a trapezoid: two parallel vertical sides of heights g(x_i) and g(x_i+1), and a horizontal width h = x_i+1 - x_i.

A single trapezoid has area h/2 · [g(x_i) + g(x_i+1)]. Adding the areas of all n trapezoids gives the composite trapezoidal approximation of the integral. The word "composite" means the interval is split into several panels instead of one.

This rule belongs to the family of numerical quadrature methods. It is valued for transparency: every step has a clear geometric meaning, and the only parameter is the subdivision count n. The main idea is easy to state: straighter, narrower panels follow the integrand more closely, so the estimate improves as n grows.

For arc length, the integrand is not the original function f(x). It is the arc-length integrand g(x) = √(1 + (f'(x))²), which measures how fast length accumulates along the curve. Keeping this distinction clear prevents the most common misunderstanding of the method.

The trapezoidal rule formula has two forms: a single-panel version and the composite version used in practice. Both come from the same trapezoid geometry.

Single-panel formula. With one trapezoid covering the whole interval:

The width (b - a) is multiplied by the average of the two endpoint heights. This is exactly the area of one trapezoid.

Composite formula. Split [a, b] into n equal subintervals of width h = (b - a)/n, with sample points x₀ = a, x₁ = a + h, ..., x_n = b:

Every symbol has a fixed meaning:

• T_n is the numerical approximation with n panels.
• h is the subinterval width, (b - a)/n.
• a and b are the lower and upper integration bounds.
• n is the number of subintervals, a positive integer.
• x_i are the sample points, spaced h apart.
• g(x_i) are the integrand values at those points.

The endpoints receive half weight because each interior sample point is shared by two neighboring trapezoids, while each endpoint belongs to only one. Algebraically, adjacent h/2 · [g(x_i) + g(x_i+1)] terms combine so that interior values count fully and the two outer values count half.

A short worked calculation. Approximate ∫₀¹ x² dx with n = 2. Then h = 0.5, x₀ = 0, x₁ = 0.5, x₂ = 1, and T₂ = 0.5 · [0/2 + 0.25 + 1/2] = 0.5 · 0.75 = 0.375. The exact integral is 1/3 ≈ 0.333, so the coarse two-panel estimate is in the right neighborhood, with room for refinement.

Arc length is an integral, so the trapezoidal rule applies to it directly. The curve y = f(x) has arc length from x = a to x = b given by:

The calculator forms the arc-length integrand g(x) = √(1 + (f'(x))²) and feeds it to the composite trapezoidal formula. The complete procedure is:

1. Start with y = f(x). Confirm the function is differentiable on [a, b].
2. Find the derivative f'(x). The tool differentiates the entered expression internally.
3. Square the derivative. Form (f'(x))². Squaring removes sign information because length accumulates regardless of slope direction.
4. Add 1. The integrand is 1 + (f'(x))², which is never smaller than 1.
5. Take the square root. This gives g(x), the local rate at which length accumulates per unit of x.
6. Divide the interval. Choose n and compute h = (b - a)/n.
7. Evaluate the integrand. Compute g(x₀), ..., g(x_n) at the sample points.
8. Apply the composite rule. Combine the values with half weights at the endpoints.
9. Interpret the result. The sum is the approximate length of the curve segment.

The derivative is squared and then placed under a square root for a geometric reason: a tiny curve segment dx wide and dy tall has length √(dx² + dy²) = √(1 + (dy/dx)²) dx by the Pythagorean theorem. Integrating that local length over [a, b] gives the total length, and the trapezoidal rule approximates the integral.

The integrand is always nonnegative because 1 + (f'(x))² is at least 1. Its minimum value is 1, reached wherever the curve is flat. Steeper slopes make g(x) larger, which is exactly why steep curves are longer than shallow ones over the same x-interval.

This fully worked example shows every number the trapezoidal rule produces. Use f(x) = x² on [0, 1] with n = 4.

1. Derivative. f'(x) = 2x.

2. Integrand. g(x) = √(1 + (2x)²) = √(1 + 4x²).

3. Step size. h = (1 - 0)/4 = 0.25.

4. Sample points. 0, 0.25, 0.5, 0.75, 1.

5. Integrand values and weights.

6. Weighted sum. 1.000000/2 + 1.118034 + 1.414214 + 1.802776 + 2.236068/2 = 5.953057.

7. Multiply by h.

8. Compare with the exact value. The exact arc length is L = √5/2 + asinh(2)/4 ≈ 1.478943. The n = 4 estimate 1.488264 exceeds it by 0.009321, an error of about 0.63%. Repeating with n = 8 gives 1.481272, already within 0.002330 of the exact value.

Notice the estimate is larger than the exact value. For this convex integrand, straight-line chords sit above the curve, so the trapezoidal rule overestimates. That systematic behavior is normal, and the overestimate shrinks as n grows.

The subdivision count n and the step size h are linked by h = (b - a)/n. Together they control how finely the integrand is sampled and how accurate the estimate is.

What n represents. It is the number of trapezoid panels, a positive integer. The number of sample points is always n + 1 because consecutive panels share their boundary points.

Why n = 1 is the single-panel rule. With n = 1, h = b - a and T₁ = h/2 · [g(a) + g(b)]. This is the crudest form of the method and rarely accurate enough.

Panel widths on [0, 1]. The table below shows how h shrinks as n grows:

Smaller panels follow the integrand more closely, so the straight-line approximation inside each panel deviates less from the true curve. That is why increasing n usually improves accuracy. The tradeoff is computational: each new panel needs one more function evaluation.

More subdivisions do not guarantee a strictly better result for every integrand. Nonsmooth or oscillatory integrands can show irregular error behavior, and floating-point rounding eventually limits gains at very large n. For smooth curves like the ones in this guide, however, the improvement is steady and predictable.

A practical strategy: start with n = 10 or n = 20, then double n until the estimate stops changing at the precision you need. This convergence check is more reliable than trusting any single value of n.

Six examples cover the range from exact cases to functions whose arc lengths have no elementary closed form. Each shows the derivative, the integrand, the substitution, and the numerical result.

Example 1: Straight line, f(x) = 2x on [0, 3]. The derivative is f'(x) = 2, a constant, so the integrand is g(x) = √(1 + 4) = √5, also constant. A single panel is exact because the integrand never bends: T₁ = 3 · √5 ≈ 6.708204. The exact length 3√5 matches. This confirms the rule: the trapezoidal rule is exact whenever the integrand is linear or constant.

Example 2: Quadratic, f(x) = x² on [0, 1], n = 4. With f'(x) = 2x and g(x) = √(1 + 4x²), the full calculation in the previous section gives L ≈ 1.488264, compared with the exact 1.478943. The small gap comes from the convex integrand; chords drawn above the curve add extra area.

Example 3: Longer quadratic, f(x) = x² on [0, 2], n = 8. Here h = 0.25 again, but the interval is twice as long and the integrand steeper. The estimate is T₈ = 4.656890. A high-resolution calculation gives 4.646784, so the error is about 0.010106. Extending the interval does not just scale the answer: the integrand grows with x, so the arc length grows faster than the interval length.

Example 4: Sine curve, f(x) = sin x on [0, π]. The derivative is f'(x) = cos x, giving g(x) = √(1 + cos²x). With n = 8, the estimate is 3.820198; with n = 16, it is 3.820198, unchanged at six decimal places. The symmetric integrand converges quickly because its curvature is gentle and evenly distributed. (Enter angles in radians; degrees would produce a different curve.)

Example 5: Logarithm, f(x) = ln(x + 1) on [0, 2], n = 10. The derivative is f'(x) = 1/(x + 1), so g(x) = √(1 + 1/(x + 1)²). The domain requires x > -1, which the interval satisfies. The estimate is T₁₀ = 2.304215, while a refined calculation gives 2.301988, an error of about 0.002227. Near x = 0 the slope is steepest, so most of the length accumulates early in the interval.

Example 6: Cubic, f(x) = x³ on [0, 1], n = 10. Here f'(x) = 3x² and g(x) = √(1 + 9x⁴). This arc length has no elementary antiderivative, which is exactly when numerical integration earns its place. The estimate is T₁₀ = 1.552609 against a refined value of 1.547866, an error of about 0.004743. Without a numerical method, this length would be very hard to obtain.

Every result above was computed directly from the composite formula, rounded to six decimal places. Repeating any of these in the calculator with the same n reproduces the same digits.

Convergence means the approximations settle toward the true value as n grows. The table below tracks f(x) = x² on [0, 1], whose exact length is 1.478943:

Each time n doubles, the error drops by roughly a factor of four: 0.037181 to 0.009321 to 0.002330 to 0.000582 to 0.000146 to 0.000036. This is second-order convergence, the signature of the composite trapezoidal rule on smooth integrands. Halving h cuts the leading error term to about one quarter.

The "change from previous" column is the practical convergence check the calculator workflow recommends. When doubling n changes the answer by less than your required precision, the estimate is stable. For this curve, n = 16 already pins the length to within 0.000582 of the truth.

The error of the composite trapezoidal rule has a precise mathematical form. For a sufficiently smooth integrand g on [a, b]:

for some ξ in [a, b]. The sign is important: when g'' is positive (convex integrand), the error is negative, meaning T_n overestimates the integral. That matches the x² example, where every estimate exceeded the exact 1.478943.

A guaranteed bound follows by replacing g''(ξ) with its maximum magnitude M = max|g''(x)| on [a, b]:

Key terms. Absolute error is |I - T_n|, the distance between truth and estimate. Relative error divides that by |I|. Truncation error is the error from approximating the integral by a finite sum, as opposed to rounding error from finite-precision arithmetic. The second derivative g'' measures the integrand's curvature, which is what the straight-line panels fail to capture.

Concrete bound for the x² example. On [0, 1], g''(x) = 4/(1 + 4x²)^3/2, largest at x = 0 with M = 4. The bound gives |I - T₄| ≤ 4/(12 · 16) = 0.020833. The actual error 0.009321 is comfortably inside it. Bounds are conservative by design; they promise a ceiling, not an exact error.

Two warnings. First, g is the arc-length integrand, not the original f. Differentiating twice through the square root is what M describes. Second, error bounds can be painful to compute for complicated functions because M itself may need maximization. In practice, the convergence check of doubling n is the more useful tool, and the calculator does not display an automatic error bound.

Simpson's Rule is the closest relative of the trapezoidal rule. Both split [a, b] into equal subintervals, but Simpson fits a parabola through three consecutive points where the trapezoidal rule draws one straight chord through two.

On smooth integrands, Simpson's fourth-order convergence usually wins: doubling n cuts its error by roughly sixteen, against four for the trapezoidal rule. But Simpson is not universally better. For rough or discontinuous integrands, both methods degrade, and the extra accuracy order may not materialize. The trapezoidal rule also accepts odd n, which is convenient for quick checks.

A sound workflow uses both: the trapezoidal estimate as a transparent baseline, and Simpson's Rule for higher precision. When the two agree to your required digits, confidence is high. When they disagree, increase n before trusting either.

Two different "trapezoid-like" ideas are easy to confuse. They produce different numbers for the same curve, so the distinction matters.

Method 1: Trapezoidal integration of the arc-length integrand. This is what the calculator does. It integrates g(x) = √(1 + (f'(x))²) with trapezoid panels drawn under the integrand curve:

Method 2: Polygonal (chord-sum) approximation of the original curve. This connects adjacent points on the curve itself with straight segments and adds their lengths:

A chord-sum joins points (x_i, f(x_i)) on the original curve. A trapezoidal panel instead approximates area under the integrand g(x). The panels do not represent straight segments of the original curve, even though both pictures involve straight lines.

Both methods converge to the true length as n grows, and both are legitimate numerical tools. The calculator on this page implements Method 1. If you need chord sums from coordinate data, the arc length from points tool follows Method 2 instead.

The trapezoidal rule is the right choice when simplicity and transparency matter more than squeezing out the last digits of accuracy.

• Learning numerical integration. Every step has a visible geometric meaning, which makes it the standard first method in calculus courses.
• Estimating curve lengths quickly. A handful of panels often gives two or three correct digits for smooth curves.
• Checking analytical solutions. Compare a hand-derived arc length against an independent numerical estimate.
• Engineering and physics models. Length estimates for cables, arches, and trajectories where approximate values feed larger calculations.
• Comparing numerical methods. It is the baseline against which Simpson's Rule and higher-order methods are judged.
• Smooth sampled functions. When f is smooth and n can be moderate, the method is dependable.
• Educational verification. Its predictable O(h²) behavior makes convergence checks easy to interpret.

Reach for a different method when the integrand changes rapidly, has kinks or discontinuities, or when high precision is required at low cost. For smooth curves, Simpson's Rule usually delivers more digits per panel. For parametric, polar, or space curves, use the dedicated parametric, polar, or 3D arc length calculators, which apply the same numerical ideas in the right coordinate system.

These are the errors that show up most often in student work and in support questions. Each is easy to avoid once you know it exists.

1. Entering incorrect integration bounds. The segment [0, 2] is not the same curve portion as [2, 0]. Always confirm a < b and that the bounds match the problem.
2. Using the wrong function. A typo like x^3 instead of x^2 changes the derivative and every downstream value.
3. Forgetting the derivative in the arc-length formula. Arc length integrates √(1 + (f')²), never f itself.
4. Applying the rule directly to f(x) for arc length. The panels must sit under g(x). Integrating f(x) computes signed area, not length.
5. Choosing too few subdivisions. n = 2 on a curved integrand can be off by several percent; the x² example had 2.5% error at n = 2.
6. Using zero subdivisions. n = 0 means h = (b - a)/0, which is undefined. n must be a positive integer.
7. Entering a noninteger subdivision count. n = 2.5 has no meaning for panel counting; round to a whole number.
8. Confusing n with the number of sample points. n panels need n + 1 sample points. With n = 4 there are 5 points, not 4.
9. Forgetting the endpoint half weights. Giving g(x₀) and g(x_n) full weight inflates the estimate by h/2 · (g(x₀) + g(x_n)).
10. Using the wrong step size. h = (b - a)/n, not b - a and not 1/n. On [0, 2] with n = 8, h = 0.25.
11. Mixing degrees and radians. sin(x) with x in degrees is a different curve than sin(x) in radians; arc lengths differ accordingly.
12. Ignoring function domain restrictions. ln(x + 1) requires x > -1. Evaluating outside the domain produces meaningless values.
13. Confusing approximation with exact integration. T_n approaches L; it equals L only for special integrands such as constants.
14. Assuming more panels always guarantee better accuracy. True in the asymptotic regime for smooth integrands, but rounding error and nonsmoothness can break the pattern at extremes.
15. Ignoring convergence checks. One estimate is a guess; two estimates at n and 2n that agree is evidence.
16. Confusing Simpson's Rule with the trapezoidal rule. Simpson uses 1-4-2-4-...-1 weights and needs even n; the trapezoidal rule uses 1/2-1-...-1-1/2 and any positive n.
17. Confusing trapezoidal integration with chord summation. Panels under the integrand are not chords on the curve; the two methods give different numbers.
18. Reporting excessive decimal precision. If the error is around 0.002, writing six decimal places overstates what is known. Round to match the verified accuracy.

The same panel arithmetic behind arc length shows up wherever a smooth quantity must be accumulated numerically.

• Engineering curve-length estimation. Lengths of cables, suspension curves, and pipe bends are computed from design functions before fabrication.
• Mechanical design. Cam profiles and gear tooth curves need accurate perimeter estimates for material planning.
• Road and pathway geometry. Alignment curves for roads and railways are measured along their centerlines, which are rarely elementary functions.
• Computer-aided design. CAD kernels estimate curve lengths for rendering, toolpath generation, and dimensioning.
• Physics simulations. Particle trajectories and field-line lengths are accumulated from numerical models.
• Numerical modeling. Any integral without a closed form, from probability densities to signal energy, can start with trapezoidal panels.
• Mathematical education. The rule is the standard gateway to numerical analysis because its error behavior is easy to demonstrate.
• Curve measurement and verification. Physical measurements of irregular curves are cross-checked against numerical predictions.

For specialized geometry, the method adapts: the same composite rule integrates parametric speed √((x')² + (y')²) and polar integrands. The calculator here handles the Cartesian case; related tools cover the other coordinate systems. It does not process CAD files or survey datasets directly; those need to be reduced to a function and bounds first.

Work these by hand, then check against the calculator. Each solution shows the substitution so you can find your own arithmetic slips.

Problem 1. Use the trapezoidal rule with n = 1 to approximate ∫₀² 3 dx, and state whether the result is exact.
Solution. h = 2, T₁ = 2/2 · [3 + 3] = 6. The integrand is constant, so the single trapezoid matches the rectangle exactly. The exact integral is also 6, so the result is exact.

Problem 2. Find the arc length of f(x) = x on [0, 1] using the trapezoidal rule with n = 1.
Solution. f'(x) = 1, so g(x) = √2, a constant. T₁ = 1 · √2 ≈ 1.414214, which is exact because the integrand is constant. Geometrically this is the diagonal of a unit square.

Problem 3. Approximate the arc length of f(x) = x² on [0, 1] with n = 2.
Solution. h = 0.5; g(0) = 1, g(0.5) = 1.414214, g(1) = 2.236068. T₂ = 0.5 · [1/2 + 1.414214 + 2.236068/2] = 0.5 · 3.032249 = 1.516124. The exact value is 1.478943, so the error is 0.037181.

Problem 4. Approximate the arc length of f(x) = sin x on [0, π] with n = 4.
Solution. g(x) = √(1 + cos²x), h = π/4. Evaluating at 0, π/4, π/2, 3π/4, π and applying the weights gives T₄ ≈ 3.819944. With n = 8 the value is 3.820198, so the n = 4 result is already within 0.000254 of the converged value.

Problem 5. Compute the step size for a = 2, b = 5, n = 6.
Solution. h = (5 - 2)/6 = 3/6 = 0.5. The sample points are 2, 2.5, 3, 3.5, 4, 4.5, 5: seven points for six panels.

Problem 6. For n = 10, how many sample points are there, and what are their weights?
Solution. There are n + 1 = 11 sample points. The endpoints x₀ and x₁₀ carry weight 1/2; the nine interior points x₁ through x₉ carry weight 1.

Problem 7. For f(x) = x² on [0, 1], compare T₂ and T₄ and state the change.
Solution. T₂ = 1.516124 and T₄ = 1.488264, a change of 0.027860. The exact value is 1.478943, so doubling n cut the error from 0.037181 to 0.009321, a factor of about four, as second-order convergence predicts.

Problem 8. T₈ = 1.481272 for f(x) = x² on [0, 1]. Report the absolute and relative error against the exact 1.478943.
Solution. Absolute error = |1.481272 - 1.478943| = 0.002330. Relative error = 0.002330 / 1.478943 ≈ 0.001576, about 0.16%. Both the estimate and the error should be quoted to three decimal places here; the extra digits in 1.481272 are not yet trustworthy.