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Integral Calculator

Compute a definite integral numerically with Simpson's Rule from a function and bounds.

Inputs stay in your browser
Browser Based
Free to use now

Definite Integral Calculator

Numerical estimate of a definite integral. Not a computer algebra system and not a symbolic antiderivative.

dx

99–9,000. Simpson 3/8 rounds n up to a multiple of 3.

Supported Syntax

  • ^ for exponents (e.g., x^2)
  • * for multiplication (e.g., 2*x or 2x)
  • / for division
  • sqrt(x) for square root
  • e and pi as constants

Functions

  • sin(x)
  • cos(x)
  • tan(x)
  • log(x) (natural)
  • log10(x)
  • abs(x)

What Is a Definite Integral Calculator?

A definite integral calculator evaluates the area under a curve between two bounds without requiring you to find the antiderivative by hand. You provide the function and the interval [a, b], and the tool computes the numerical result using a high-accuracy approximation method.

This page uses Simpson's 3/8 Rule with a high subdivision count to approximate the integral to several decimal places. Tool inputs stay in the browser.

How to Use the Integral Calculator

1
Enter your function. Type your function using standard math notation. Supported syntax includes x^2, sin(x), cos(x), e^x, ln(x), sqrt(x), and combinations thereof.
2
Set the lower bound (a). Enter the lower limit of integration — the starting x-value for the interval.
3
Set the upper bound (b). Enter the upper limit of integration — the ending x-value for the interval.
4
Calculate and read the result. The engine evaluates the integral using Simpson's 3/8 Rule with a high subdivision count and returns the numerical result.

Who Is This For?

  • →Calculus students checking homework answers or verifying hand-computed definite integrals before submitting problem sets.
  • →Engineering and physics students computing areas, volumes, work, or accumulated quantities where the antiderivative exists but the algebra is error-prone.
  • →Anyone who needs to evaluate a definite integral quickly without installing a computer algebra system or relying on a server-side CAS.

Key Benefits

  • ✓Privacy. Tool inputs stay in the browser.
  • ✓Free to use now. No account, no paywall, no CAS license required. Ads are not live yet.
  • ✓No account needed. Works instantly — open the page and start calculating.
  • ✓High-accuracy numerical method. Uses Simpson's 3/8 Rule with thousands of subdivisions for decimal-level precision across most smooth functions.

Common Integration Mistakes

  • ✗Forgetting to set correct bounds — a definite integral from 0 to 1 gives a completely different result than 0 to 10; always verify your interval before interpreting the output.
  • ✗Using numerical integration on discontinuous functions — if your function has a vertical asymptote inside the interval, the result will be inaccurate; split the integral at the discontinuity.
  • ✗Confusing indefinite and definite integrals — this tool computes a specific numerical area; it does not return the symbolic antiderivative with a constant C.

Integration is often paired with other mathematical tools — the Fraction Calculator helps verify exact rational results, and the Unit Converter is useful when working with physical quantities that need unit normalization before integration.

For symbolic antiderivatives and step-by-step integral solutions, see Wolfram Alpha's Integral Calculator, which complements this numerical tool for problems requiring closed-form answers.

Common Use Cases

Homework verification. Check a definite integral result computed by hand before submitting it. If your answer and the calculator diverge, you know where to look for an algebra error.

Area between curves. In engineering and physics, compute the net area between two functions over an interval by entering f(x) − g(x) as the integrand.

Accumulated quantities. Evaluate integrals of rate functions — displacement from velocity, total energy from power, accumulated charge from current — quickly without symbolic manipulation.

Simpson's 3/8 Rule

To calculate the area under the curve, this tool implements Simpson's 3/8 Rule. This is a highly accurate method for numerical integration that approximates the required function using cubic parabolas rather than standard straight lines or flat rectangles (like the Riemann sum).

Under the hood, we divide your interval [a, b] into thousands of microscopic sub-intervals so the approximation usually lands within several decimal places for smooth functions — discontinuous or poorly behaved integrands can still miss.

Runs in your browser

The integration engine runs in this tab. Tool inputs stay in the browser. The site uses aggregate Google Analytics.

Frequently Asked Questions

What is an integral? ↓
An integral computes the area under a curve. A definite integral gives a number representing the net area between the curve and the x-axis within two specified bounds. An indefinite integral gives the antiderivative function — the general formula whose derivative equals the original function. This tool evaluates definite integrals numerically.
Is this integral calculator free? ↓
Yes. This Integral Calculator is free to use now — ads are not live yet — with no account required. Tool inputs stay in your browser. The site uses aggregate Google Analytics.
What is the difference between a definite and indefinite integral? ↓
A definite integral has upper and lower bounds (a and b) and returns a single number — the net area under the curve between those two points. An indefinite integral has no bounds and returns a function plus an arbitrary constant C, representing the general antiderivative. This calculator evaluates definite integrals only.
What is the Fundamental Theorem of Calculus? ↓
The Fundamental Theorem of Calculus states that differentiation and integration are inverse operations. Specifically, the definite integral of f(x) from a to b equals F(b) − F(a), where F is any antiderivative of f. This theorem is why computing a definite integral symbolically requires finding the antiderivative first, then evaluating it at the bounds.
Does this solve indefinite integrals? ↓
No. This tool is designed strictly for evaluating definite integrals — approximating the numerical area between two defined bounds, a and b, with Simpson's 3/8 Rule. It does not perform symbolic algebra to return an antiderivative function with +C. For indefinite integrals, you need a computer algebra system (CAS).
What happens if the function is undefined at a point in the interval? ↓
If you attempt to integrate over a vertical asymptote — for example, evaluating 1/x across an interval that includes 0 — the numerical engine will detect the divergence and return an error. The integral is undefined or divergent at that point, and no finite result can be computed.
Disclaimer

This tool is provided for informational and educational purposes only. Results are not a substitute for professional advice.

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