Simpson's Rule Calculator
Approximate a definite integral using composite Simpson's 1/3 rule with n subintervals.
Inputs
Integrand using x.
Approximate integral
2.0001095173
Step size h
0.31415927
Step by step
Values used
f(x) = sin(x); Lower bound (a) = 0; Upper bound (b) = 3.14; Subintervals (n, must be even) = 10
Simpson's 1/3 rule
∫[a,b]f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]
Approximate integral
= 2.0001095173
Step size h
= 0.31415927
How it works
Simpson's 1/3 rule approximates ∫f(x)dx by fitting quadratic polynomials through consecutive triplets of points. The composite version splits [a,b] into n equal subintervals (n must be even) and applies the basic rule to each pair. Error is O(h⁴), making it much more accurate than the trapezoidal rule for the same n.
Formula
Simpson's 1/3 rule
∫[a,b]f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]
- h
- (b-a)/n
- n
- Number of subintervals (even)
Frequently Asked Questions
Why must n be even?
Simpson's rule uses pairs of subintervals (triplets of points) to fit parabolas. An odd n would leave an unpaired subinterval.
How accurate is Simpson's rule?
The error is proportional to h⁴·f''''(c) for some c in [a,b]. Doubling n reduces error by a factor of 16.