Definite integral calculator
Compute the area under a curve between two limits using Simpson’s rule.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Compute the area under a curve between two limits using Simpson’s rule.
- Formula used
∫ₐᵇ f(x) dx ≈ (h/3) · [f(a) + 4·Σf(odd) + 2·Σf(even) + f(b)]- Example
- ∫₀³ (x² + 1) dx
Definite integral calculator
This calculator solves definite integrals, that is, the numeric value of the area between the function and the X axis over an interval.
It does not return the symbolic antiderivative: it returns the number, which is what most practical applications need.
How it works
Composite Simpson's rule is applied, approximating the function with parabolas over each pair of subintervals. With 1000 subintervals the error is usually negligible for smooth functions.
Where the function dips below the X axis, that area subtracts.
Formula
∫ₐᵇ f(x) dx ≈ (h/3) · [f(a) + 4·Σf(odd) + 2·Σf(even) + f(b)]
with h = (b − a) / n and n even
Mean value: (1/(b−a)) · ∫ₐᵇ f(x) dx
Worked example
∫₀³ (x² + 1) dx
Antiderivative: x³/3 + x
Evaluated: (27/3 + 3) − 0 = 12
Simpson's rule returns 12.00000000
Explanation
An integral is accumulated area
Adding infinitely many infinitely narrow rectangles under a curve gives its area. That idea, which sounds like a game, turns out to be the inverse of differentiation, and that connection is the fundamental theorem of calculus: one of the most important results in the history of mathematics.
What gets accumulated in practice
If the curve represents speed, the area under it is distance travelled. If it represents flow rate, it is total volume. If it represents power, it is energy consumed. Any quantity obtained by accumulating another over time is an integral.
Why it is computed by approximation
Many perfectly ordinary functions have no antiderivative expressible in elementary functions. The Gaussian bell curve is the famous example: the whole of statistics depends on its integral, and it must be computed numerically. Hence methods like Simpson's rule, which approximates the curve with parabolas.
Signed area
Anything below the horizontal axis counts as negative. If you want total geometric area and the function crosses the axis, you must separate the sections and add the absolute values, or the result will come out smaller than expected.
Frequently asked questions
Why must the number of subintervals be even?
Simpson’s rule pairs points to fit a parabola on each segment. If you enter an odd number it is adjusted automatically.
Does it handle improper integrals?
No. The limits must be finite and the function must be defined across the whole interval.
Why is my integral negative?
Because the function dips below the horizontal axis over part of the interval. The integral counts that area as negative; for geometric area, split the sections.
How are differentiation and integration related?
They are inverse operations. The fundamental theorem of calculus states that integrating a function and then differentiating the result returns the original function.
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