# Integral Calculator

Calculate definite integrals numerically using Simpson's rule. Supports polynomial, trigonometric, exponential, and logarithmic functions with error estimation.

## What this calculates

Compute definite integrals of common functions using numerical integration. This calculator uses Simpson's 1/3 rule to approximate the area under a curve between two bounds, and provides an error estimate so you know how accurate the result is.

## Inputs

- **Function Type** — options: Polynomial: ax^n + bx + c, Trigonometric: a·trig(bx), Exponential: a·e^(bx), Logarithmic: a·ln(bx) — Choose the type of function to integrate.
- **Trig Function** — options: sin, cos, tan — Only used for trigonometric function type.
- **Coefficient a** — Leading coefficient. Polynomial: a in ax^n + bx + c. Trig: a in a·trig(bx). Exponential: a in a·e^(bx). Log: a in a·ln(bx).
- **Coefficient b** — Second coefficient. Polynomial: b in ax^n + bx + c. Trig/Exp/Log: b (frequency or rate parameter).
- **Constant c (polynomial only)** — Constant term in the polynomial ax^n + bx + c.
- **Exponent n (polynomial only)** — The power of x in the polynomial term ax^n.
- **Lower Bound (a)** — The lower limit of the definite integral.
- **Upper Bound (b)** — The upper limit of the definite integral.
- **Number of Intervals** — min 2, max 100000 — More intervals give higher accuracy. Must be even for Simpson's rule. Default is 1000.

## Outputs

- **Definite Integral Result** — The numerical value of the definite integral.
- **Integral Expression** — formatted as text — The integral that was evaluated.
- **Absolute Error Estimate** — Estimated error by comparing results with n and n/2 intervals.

## Details

Integration finds the total accumulated value of a function over an interval. Geometrically, a definite integral represents the signed area between a curve and the x-axis from point a to point b.

**How Simpson's Rule Works:**

Simpson's 1/3 rule approximates the integral by fitting parabolas through consecutive groups of three points along the curve, then summing the areas under those parabolas. It is significantly more accurate than the trapezoidal rule for smooth functions.

The formula splits the interval [a, b] into n equal subintervals (n must be even), then computes:

∫f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]

where h = (b - a)/n.

**Supported Function Types:**

- **Polynomial (ax^n + bx + c):** Works with any real exponent. For example, integrating x^2 from 0 to 1 gives 1/3.
- **Trigonometric (a·sin(bx), a·cos(bx), a·tan(bx)):** Useful for wave and oscillation problems. Integrating sin(x) from 0 to π gives exactly 2.
- **Exponential (a·e^(bx)):** Models growth and decay. Integrating e^x from 0 to 1 gives e - 1 ≈ 1.71828.
- **Logarithmic (a·ln(bx)):** Make sure bx stays positive over the entire interval, since ln is only defined for positive arguments.

**Error Estimation:**

The calculator runs Simpson's rule twice, once with your chosen number of intervals and once with half as many. The difference between those two results gives a practical estimate of the approximation error. Increasing the number of intervals generally reduces the error.

## Frequently Asked Questions

**Q: What is a definite integral?**

A: A definite integral calculates the net signed area between a function's curve and the x-axis over a specific interval [a, b]. Regions above the x-axis count as positive area, and regions below count as negative. For example, the integral of x^2 from 0 to 3 equals 9, which is the area under the parabola between x = 0 and x = 3.

**Q: How accurate is Simpson's rule?**

A: Simpson's rule is very accurate for smooth functions. Its error decreases proportionally to h^4 (where h is the subinterval width), so doubling the number of intervals makes the result roughly 16 times more accurate. For most practical calculations, 1000 intervals gives results accurate to 8 or more decimal places.

**Q: Why must the number of intervals be even?**

A: Simpson's rule works by fitting a parabola through every group of three consecutive points. Each parabola covers two subintervals, so the total number of subintervals must be even. If you enter an odd number, the calculator automatically rounds up to the next even number.

**Q: Can this calculator handle improper integrals or infinite bounds?**

A: This calculator only handles proper definite integrals with finite bounds. If the function has a singularity (like 1/x at x = 0) within the interval, or if you need to integrate to infinity, you would need to apply a limit or substitution technique first, then use this tool on the transformed integral.

**Q: What is the difference between numerical and analytical integration?**

A: Analytical integration finds an exact symbolic antiderivative (like ∫x^2 dx = x^3/3 + C). Numerical integration, which this calculator uses, approximates the definite integral's value by sampling the function at many points. Numerical methods work for functions that are difficult or impossible to integrate analytically, at the cost of a small approximation error.

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Source: https://vastcalc.com/calculators/math/integral
Category: Math
Last updated: 2026-04-08