Click to show tips
Pick a scenario to see how the calculator works, then adjust the values
Integrate x^2 from 0 to 1 using Simpson's rule. Exact answer: 1/3.
Key values: x^2 · [0, 1] · Simpson's rule
Integrate sin(x) from 0 to pi. Exact answer: 2.
Key values: sin(x) · [0, pi] · Exact: 2
Approximate the Gaussian integral e^(-x^2) from -3 to 3.
Key values: e^(-x^2) · [-3, 3] · Gaussian
Enter a function f(x) with bounds and click Calculate to see the numerical integral, error estimate, and interactive Riemann sum visualization.
The trapezoidal rule approximates the area under a curve by dividing the interval into trapezoids rather than rectangles. Each trapezoid connects consecutive function values with a straight line, capturing the slope of the function.
where . Unlike Simpson's rule, there is no restriction on being even.
The error depends on the second derivative. This means the trapezoidal rule is exact for linear functions (whose second derivative is zero). Doubling reduces the error by a factor of 4.
The trapezoidal rule approximates the area under a curve by dividing the interval into trapezoids. Each trapezoid connects consecutive function values with a straight line, providing convergence.
The trapezoidal rule has convergence while Simpson's rule has . However, the trapezoidal rule has no parity constraint on the number of subintervals and excels for periodic functions integrated over a full period.
Yes, the trapezoidal rule is exact for linear functions because its error bound depends on the second derivative, which is zero for first-degree polynomials.
The trapezoidal rule converges exponentially fast for smooth periodic functions integrated over a complete period. It is also the natural choice when you have experimental data at fixed measurement points rather than a formula.
Doubling (halving ) reduces the trapezoidal rule error by a factor of 4, since the error is proportional to . For example, going from 100 to 200 subintervals cuts the error to one-quarter.