How to Use This Calculator
Calculate the definite integral of f(x) = x² from 0 to 2. Enter "x^2" as the function, set Lower Bound a = 0, Upper Bound b = 2, Subintervals n = 100, and click Calculate. The result is approximately 2.666667. The exact answer is 2³/3 = 8/3 = 2.666667, so Simpson's rule with 100 subintervals gives near-perfect accuracy for a quadratic function. The step-by-step shows: h = (2-0)/100 = 0.02, then applies the Simpson's formula. Now try f(x) = sin(x) from 0 to π. Enter "sin(x)", a = 0, b = 3.14159, n = 100. The result should be about 2.0 — which is the exact integral of sin(x) from 0 to π, since the antiderivative -cos(x) evaluated from 0 to π gives -(-1) - (-1) = 2.
About Integral Calculator
This integral calculator computes definite integrals numerically using Simpson's rule, one of the most accurate numerical integration methods. Simply enter your function f(x), the lower bound a, the upper bound b, and the number of subintervals n (higher values yield greater accuracy). The calculator evaluates the function at multiple points within the interval and applies Simpson's composite rule to approximate the area under the curve. Results include the numerical value, bounds, method details, and estimated error. This tool is essential for calculus students verifying analytic solutions, engineers performing quick calculations, and scientists analyzing accumulated quantities. The step-by-step breakdown shows the formula application for educational understanding.
When to Use This Calculator
Physics (Work & Energy): The work done by a variable force F(x) from position a to b is the integral of F(x)dx. If a spring with force F(x) = 50x N (Hooke's law) is stretched from 0 to 0.5 meters, the work is ∫₀⁰·⁵ 50x dx = 6.25 Joules. Enter "50*x", a = 0, b = 0.5, n = 100 to verify.
Probability & Statistics: The area under a probability density function between two points gives the probability of a value falling in that range. The standard normal distribution requires numerical integration since it has no closed-form antiderivative. Enter the PDF formula and integrate between z = -1 and z = 1 to find that about 68% of data falls within ±1 standard deviation.
Engineering (Area & Volume): The volume of a solid of revolution can be computed by integrating the cross-sectional area. For a vase with radius r(x) = 2 + 0.1x² from x = 0 to x = 10, the volume is ∫ π × (2 + 0.1x²)² dx. Use the integral calculator with function "pi*(2+0.1*x^2)^2" to find the volume in cubic units.