Definite Integral Calculator – Compute Integrals Numerically
Calculate definite integrals numerically with our free online integral calculator. Evaluate the area under any function curve over any interval with accurate numerical results using Simpson's Rule.
Supports: x^2, sin(x), cos(x), exp(x), ln(x), sqrt(x), etc.
Higher values give more accurate results (max 10000)
Understanding Definite Integrals
A definite integral calculates the area under a curve between two points. It's one of the two main operations in calculus (the other being differentiation), connected by the Fundamental Theorem of Calculus.
What Is a Definite Integral?
∫[a to b] f(x) dx = Area under f(x) from x=a to x=bThe integral adds up infinitely many infinitely thin rectangles under the curve. The result tells you the net signed area — regions above the x-axis count positive, regions below count negative.
Simpson's Rule for Numerical Integration
Not all integrals can be solved with a neat formula. For those, we use numerical methods. This calculator uses Simpson's Rule, which approximates the curve with parabolas instead of rectangles.
∫f(x)dx ≈ (h/3) × [f(a) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(b)]The pattern 1, 4, 2, 4, 2, ..., 4, 1 weights the function values. More intervals mean better accuracy.
Worked Examples
Example 1: ∫x² dx from 0 to 1
The exact answer is 1/3 ≈ 0.333...
∫[0 to 1] x² dx = [x³/3][0 to 1] = 1/3 - 0 = 0.333...This represents the area under the parabola y = x² from x=0 to x=1.
Example 2: ∫sin(x) dx from 0 to π
The area under one hump of the sine curve.
∫[0 to π] sin(x) dx = [-cos(x)][0 to π] = -(-1) - (-1) = 2The result is exactly 2 — the area of the region bounded by y=sin(x) and the x-axis.
Example 3: ∫(1/x) dx from 1 to 2
This gives us ln(2).
∫[1 to 2] (1/x) dx = [ln(x)][1 to 2] = ln(2) - ln(1) ≈ 0.693The natural logarithm is defined as this integral.
A Quick Fact
The integral symbol ∫ was introduced by Gottfried Wilhelm Leibniz in 1675. It's an elongated "S" standing for "summa" (Latin for sum) — because integration is fundamentally about summing infinitely many infinitesimal pieces.
Common Questions
Why use numerical integration instead of exact formulas?
Many functions don't have elementary antiderivatives. Functions like e^(-x²), sin(x)/x, or 1/ln(x) can't be integrated in closed form. Numerical methods work for any continuous function.
How accurate is Simpson's Rule?
The error is proportional to 1/n⁴, where n is the number of intervals. Doubling the intervals reduces error by about 16x. With 1000 intervals, you typically get 6+ decimal places of accuracy.
What if the function goes below the x-axis?
The integral gives the net signed area. Regions below the axis subtract from the total. If you want total area (ignoring signs), you'd integrate the absolute value.
Can I integrate functions with discontinuities?
Not directly with this tool. Improper integrals (with infinite discontinuities or infinite bounds) require special handling. The numerical method may fail or give incorrect results near singularities.
Where do definite integrals appear in real applications?
Everywhere in physics and engineering: computing work done by a variable force, finding centers of mass, calculating probabilities from density functions, determining total growth from a rate.
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