Definite Integral Calculator — Area Under a Curve
DIRECT ANSWER: this tool computes the definite integral of f(x) between two limits using numeric integration (Simpson's rule). It returns a numeric value for the given bounds — the signed area under the curve — not a symbolic antiderivative.
Calculate a definite integral
Use x as the variable. Supports + - * / ^ and sin, cos, tan, sqrt, exp, log (natural), pi.
How this calculator works
This is a numeric definite integral calculator. You give it a function f(x) and two limits, a (lower) and b (upper), and it computes the value of:
∫ab f(x) dx
That value is the signed area between the curve y = f(x) and the x-axis, from x = a to x = b. Area below the axis counts as negative.
Under the hood it uses Simpson's rule, a standard numerical-integration method. The interval [a, b] is split into n = 1000 equal strips of width h = (b − a)/n, and each pair of strips is approximated by a parabola:
∫ f(x) dx ≈ (h/3) × [ f(a) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + 4f(xn-1) + f(b) ]
With 1000 strips this is highly accurate for smooth functions. Because the method evaluates f(x) at many points, it does not produce a formula — it produces a single number for the bounds you enter.
Definite vs. indefinite integrals
- Definite integral ∫ab f(x) dx — has limits and evaluates to a number. This is what this tool returns.
- Indefinite integral ∫ f(x) dx — has no limits and evaluates to a function (the antiderivative) plus a constant C. This tool does not compute that.
Integration rules
For reference, here are common antiderivatives. The definite integral over [a, b] is found by evaluating the antiderivative F at the limits: ∫ab f(x) dx = F(b) − F(a).
| Function f(x) | Antiderivative F(x) |
|---|---|
| xn (n ≠ −1) | xn+1 / (n + 1) + C (power rule) |
| 1 / x | ln|x| + C |
| ex | ex + C |
| cos x | sin x + C |
| sin x | −cos x + C |
| sec2 x | tan x + C |
| k (constant) | kx + C |
Worked examples
Example 1 — power rule. ∫02 x² dx. Using the power rule, F(x) = x³/3, so the answer is 2³/3 − 0³/3 = 8/3 ≈ 2.6667. The calculator returns the same value numerically.
Example 2 — trig. ∫0π sin(x) dx = [−cos x]0π = −cos(π) + cos(0) = 1 + 1 = 2. Enter sin(x) with a = 0 and b = pi.
Example 3 — exponential. ∫01 ex dx = e¹ − e⁰ = e − 1 ≈ 1.71828. Enter exp(x) with a = 0 and b = 1.
Frequently asked questions
Does this calculator give the antiderivative or a symbolic answer?
No. This is a numeric definite integral calculator. It returns a single number — the value of the integral between your two limits — using Simpson's rule. It does not return a symbolic antiderivative or a formula with "+ C". If you need the symbolic indefinite integral, solve it by hand using the integration rules above or use a computer-algebra system.
How accurate is the result?
Very accurate for smooth, well-behaved functions. Simpson's rule with 1000 strips typically matches the exact value to around 6 significant figures. Accuracy can drop for functions with sharp spikes, discontinuities, or vertical asymptotes inside the interval.
Why did I get "function undefined on this interval"?
That message appears when f(x) returns an undefined or infinite value at one of the sample points — for example dividing by zero, taking log of a non-positive number, or sqrt of a negative number somewhere between a and b. Check that your function is defined across the whole interval.
What does a negative answer mean?
The definite integral is a signed area. Where the curve is below the x-axis, that region contributes a negative amount. So the result can be negative or zero even when there is visible area, if positive and negative regions cancel.
Can I swap the limits (a greater than b)?
Yes. If the lower limit a is greater than the upper limit b, the calculator follows the standard convention ∫ab = −∫ba and negates the result automatically.
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Last reviewed: June 2026