Trapezoidal & Simpson's Rule Integration

Numerically integrate f(x) with both the trapezoidal and Simpson's rules, then compare the two approximations.

Input

Enter the integrand f(x), the interval [a, b] and the number of subintervals n to approximate the definite integral with both the trapezoidal and Simpson''s rules.

Integrand f(x)

e.g. sin(x), x^2, exp(-x^2), 2x+1. You can use pi, e and functions like sin, cos, exp, log.

Lower bound a

Upper bound b

Subintervals n

An even value is used for Simpson''s rule (odd inputs are rounded to the nearest even number).

Result

Simpson''s rule approximation

2.0000000108

∫ sin(x) dx over [0, 3.1415926536]

Trapezoidal rule

1.9998355039

Simpson''s rule

2.0000000108

Difference

0.0001645069

Approximation by method

Method	Value
Trapezoidal rule	1.9998355039
Simpson''s rule	2.0000000108

Computation parameters

Subintervals n (used)	100
Step size h	0.0314159265

How it works

The trapezoidal rule splits [a, b] into n equal subintervals and approximates each strip as a trapezoid. With step h=(b-a)/n, I≈h×(f0/2+f1+…+f(n-1)+fn/2).
Simpson's rule (the 1/3 rule) approximates each pair of subintervals by a parabola: I≈(h/3)×(f0+4f1+2f2+4f3+…+4f(n-1)+fn). Because n must be even, an odd input is rounded to the nearest even value.
For smooth functions Simpson's rule is generally more accurate and integrates cubic polynomials exactly. The difference between the two results is a rough indicator of the approximation error.
The integrand supports +, -, *, /, ^ (power), parentheses, unary minus and implicit multiplication, with the constants pi and e and the functions sin, cos, tan, asin, acos, atan, sinh, cosh, tanh, exp, log (=ln), log10, sqrt, cbrt and abs.
The bounds a and b may also be expressions containing pi or e. All computation runs in your browser with double-precision floating point and nothing is sent to a server.

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