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Home › Math & Conversions › Core Math & Algebra › Trapezoidal Rule Calculator Trapezoidal Rule Calculator – Numerical Integration Approximate definite integrals using the trapezoidal rule. Work from a function \(f(x)\) on an equally spaced grid or from tabulated data points \((x_i, f(x_i))\). Designed for calculus students, instructors, and engineers who need a transparent, step-by-step numerical integration tool. Trapezoidal rule calculator Composite & data-based trapezoidal rule Choose the calculation mode, fill in the inputs, and click Calculate . The tool reports the trapezoidal approximation, effective step size, and a detailed grid table. Calculation mode Function-based (equally spaced grid) Data points \((x_i, f(x_i))\) Lower limit a Upper limit b Number of subintervals n n ≥ 1. A larger n gives a finer grid and usually better accuracy. Function \(f(x)\) Use standard notation. The calculator automatically maps sin, cos, tan, exp, ln, log, sqrt, abs, pi, e to JavaScript’s Math.* functions and interprets ^ as a power operator. Enter pairs \((x_i, f(x_i))\). Empty rows are ignored. The tool sorts your points by x and applies the basic trapezoid formula on each interval. Number of data points: Update table Calculate Clear Numerical tolerance for diagnostics: 1e-12 The trapezoidal approximation and grid table will appear here after you click Calculate . Understanding the trapezoidal rule The trapezoidal rule is one of the simplest and most widely used numerical integration formulas. Instead of computing the exact area under the curve \(y = f(x)\), you approximate the graph with straight-line segments and sum the areas of the resulting trapezoids. 1. Basic trapezoid formula On a single interval \([x_i, x_{i+1}]\) with function values \(f(x_i)\) and \(f(x_{i+1})\), the area of the trapezoid is: \[ \int_{x_i}^{x_{i+1}} f(x)\,dx \approx \frac{x_{i+1} - x_i}{2}\,\bigl(f(x_i) + f(x_{i+1})\bigr). \] This formula works whether your data come from a function or from measurements. The only requirement is that you know the x-coordinates and the corresponding function (or measured) values. 2. Composite trapezoidal rule on an equally spaced grid If you divide \([a, b]\) into \(n\) equal subintervals of width \(h = \frac{b-a}{n}\), with grid points \(x_0 = a, x_1, \dots, x_n = b\), the composite trapezoidal rule is: \[ \int_a^b f(x)\,dx \approx \frac{h}{2} \left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]. \] The function-based mode of this calculator builds exactly this grid, evaluates \(f(x_i)\) at each point, and applies the formula. 3. Error behavior If \(f\) has a continuous second derivative on \([a, b]\), the global error of the composite trapezoidal rule satisfies \[ \text{Error} \approx -\frac{(b - a)}{12} h^2\, f''(\xi) \quad \text{for some} \;\xi \in (a, b). \] This shows a quadratic dependence on the step size \(h\): halving \(h\) (doubling the number of subintervals) typically reduces the error by roughly a factor of 4, provided \(f\) is smooth enough and does not oscillate too wildly. 4. Trapezoidal rule vs. Simpson’s rule Simpson’s rule uses quadratic interpolation on pairs of subintervals and generally achieves higher accuracy (error proportional to \(h^4\)) for smooth integrands. The trapezoidal rule, however, has important advantages: It works directly with tabulated data and nonuniform x-spacing. It is easy to implement and explain in a classroom setting. It is often sufficient for engineering estimates and quick checks. In practice, you can use the trapezoidal rule to get a first estimate, then refine or cross-check with Simpson’s rule when it is available. Trapezoidal rule – FAQ How should I enter a function like sin(x) or e −x² ? + You can type expressions such as sin(x) , cos(x) , exp(-x^2) , or sqrt(1 + x^2) . The calculator automatically converts common functions to JavaScript’s Math.* equivalents and interprets ^ as a power operator. If needed, you can always write the expression explicitly, for example Math.sin(x) or Math.exp(-x*x) . What happens if the function has a discontinuity inside [a, b]? + The trapezoidal rule assumes that the function is reasonably well-behaved on the interval. If there is a jump or vertical asymptote, the approximation may be seriously misleading. In such cases you should split the integral at the discontinuity (if the improper integral makes sense) or use a method tailored for improper integrals and compare results carefully. Can this tool tell me the exact integration error? + No. The exact error depends on the second derivative of the function, which is generally unknown when you use numerical integration. The tool reports step size and grid, so you can repeat the computation with a finer grid and compare values. The difference between two successive refinements is often a good indicator of the remaining error in practical work. Core Math & Algebra tools Set Operations Arithmetic Sequence Linear Regression Modular Arithmetic Equation Solver Circular Segment Area Trig Identity Explorer Hyperbolic Function Explorer Linear Algebra Toolbox Trapezoidal Rule Calculator (you are here) Perfect Number Checker Chinese Remainder Theorem Number theory & calculus helpers Set Theory Basics Base Converter Interval Notation Helper Pythagorean Theorem Convolution Explorer Dice Roll Probability Professional use note This trapezoidal rule calculator is suited for teaching, study, and quick engineering estimates. For safety-critical or regulatory calculations, repeat the computations with independent numerical software and verify that results are stable with respect to step size and data quality.
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