Trapezoidal Rule Calculator
Approximate a definite integral using the composite trapezoidal rule with n subintervals.
Inputs
Integrand using x.
Approximate integral
0.7462107961
Step size h
0.10000000
Step by step
Values used
f(x) = exp(-x^2); Lower bound (a) = 0; Upper bound (b) = 1; Subintervals (n) = 10
Trapezoidal rule
∫[a,b]f(x)dx ≈ h·[f(x₀)/2 + f(x₁) + f(x₂) + ... + f(xₙ)/2]
Approximate integral
= 0.7462107961
Step size h
= 0.10000000
How it works
The composite trapezoidal rule approximates ∫f(x)dx by connecting adjacent points with straight lines and summing the trapezoid areas. Error is O(h²), making it less accurate than Simpson's rule but simpler and applicable with any number of subintervals.
Formula
Trapezoidal rule
∫[a,b]f(x)dx ≈ h·[f(x₀)/2 + f(x₁) + f(x₂) + ... + f(xₙ)/2]
- h
- (b-a)/n
- n
- Number of subintervals
Frequently Asked Questions
When is the trapezoidal rule preferred?
For periodic functions integrated over a full period, the trapezoidal rule can be exponentially accurate (superconvergence), outperforming even Simpson's rule.
How does error decrease with n?
Doubling n reduces error by a factor of 4 (quadratic convergence in n).