Simpson’s Rule Calculator
Accurately approximate definite integrals using Simpson’s 1/3 and 3/8 rules
Use JavaScript math functions: sin(x), cos(x), log(x), exp(x), sqrt(x), etc.
Must be even for Simpson’s 1/3 Rule, multiple of 3 for 3/8 Rule
Important Note
Simpson’s Rule provides an approximation of the definite integral. For Simpson’s 1/3 Rule, the number of intervals (n) must be even. For Simpson’s 3/8 Rule, n must be a multiple of 3.
Example Functions
Click on any example to load it into the calculator:
f(x) = x², from 0 to 2, n=4
Exact: 8/3 ≈ 2.6667
f(x) = sin(x), from 0 to π, n=6
Exact: 2
f(x) = e^x, from 0 to 1, n=4
Exact: e-1 ≈ 1.7183
f(x) = 1/x, from 1 to 2, n=6
Exact: ln(2) ≈ 0.6931
What is Simpson’s Rule?
Simpson’s Rule is a numerical method for approximating definite integrals. It provides a more accurate approximation than simpler methods like the Trapezoidal Rule by using quadratic (Simpson’s 1/3 Rule) or cubic (Simpson’s 3/8 Rule) polynomials to approximate the function over each subinterval.
How to Use This Simpson’s Rule Calculator
Our Simpson’s Rule calculator makes numerical integration simple:
- Enter the function you want to integrate using standard mathematical notation
- Specify the lower and upper limits of integration
- Choose the number of intervals (must be even for 1/3 Rule, multiple of 3 for 3/8 Rule)
- Click “Calculate Integral” to see the approximation
Applications of Simpson’s Rule
Simpson’s Rule is widely used in engineering, physics, and mathematics for approximating definite integrals when an analytical solution is difficult or impossible to find. Common applications include calculating areas, volumes, and solving differential equations.