Integral Calculator

Calculate definite integrals numerically in your browser. Uses Simpson's 3/8 Rule for high-accuracy results. Free, private, no account needed.

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Definite Integral Calculator

Evaluate definite integrals numerically directly in your browser. Supports trigonometric functions, logarithms, and standard algebraic notation.

∫

Upper Bound (b)

f(x) = dx

Lower Bound (a)

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)

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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 tool uses Simpson's 3/8 Rule with a high subdivision count to approximate the integral to several decimal places of precision. Because the computation runs entirely in your browser, your equations are never transmitted to a server — making it safe for proprietary engineering formulas and offline academic work.

How to Use the Integral Calculator

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.

Set the lower bound (a). Enter the lower limit of integration — the starting x-value for the interval.

Set the upper bound (b). Enter the upper limit of integration — the ending x-value for the interval.

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

✓Runs in your browser. Equations are never transmitted over the internet or saved to a database — safe for proprietary formulas.
✓Completely free. No subscription, no paywall, no CAS license required.
✓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 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 to ensure precision up to several decimal places.

Safe & Private

Because the integration engine runs directly in the client (your device), your equations are never transmitted over the internet or saved to a database. You can safely calculate proprietary engineering formulas or offline academic problems without worrying about data tracking.

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, completely free. It runs entirely in your browser — no account required, no data stored, and no CAS license needed. Your equations never leave your device.

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 — finding the exact numerical area between two defined bounds, a and b. 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

The tools and calculators provided on The Simple Toolbox are intended for educational and informational purposes only. They do not constitute financial, legal, tax, or professional advice. While we strive to keep calculations accurate, numbers are based on user inputs and standard assumptions that may not apply to your specific situation. Always consult with a certified professional (such as a CPA, financial advisor, or attorney) before making significant financial or business decisions.

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