About the Integration Calculator
What is the Integration Calculator?
A definite integral ∫ₐᵇ f(x) dx measures the signed area between the graph of f and the x-axis from x = a to x = b: area above the axis counts as positive and area below as negative. Integrals give total distance from speed, total work from force and total quantity from a rate.
Enter a function of x and the lower and upper limits. The calculator uses adaptive Simpson's rule, which keeps subdividing the interval wherever the function is curvy until the answer is accurate to about 10 decimal places, and also reports the average value of the function over the interval.
How to use the Integration Calculator
- Function f(x) — use x as the variable, for example x^2 + 1 or exp(-x^2). (sample: x^2 + 1)
- Lower Limit (a) (sample: 0)
- Upper Limit (b) — if b is less than a, the sign of the answer flips. (sample: 3)
- Press Calculate Integral (results also refresh as you type). The result card shows the Definite Integral, along with Average value of f on [a, b], Interval width (b − a) and Method.
Formula used
∫ₐᵇ f(x) dx ≈ (b − a) ÷ 6 × [f(a) + 4f((a + b) ÷ 2) + f(b)]
- a, b = the lower and upper limits
- f(x) = the function being integrated
This Simpson estimate is applied to smaller and smaller pieces until neighbouring estimates agree. The average value of f on [a, b] is (1 ÷ (b − a)) × ∫ₐᵇ f(x) dx.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| Function f(x) | x^2 + 1 |
| Lower Limit (a) | 0 |
| Upper Limit (b) | 3 |
| Result | Value |
|---|---|
| Definite Integral | 12 |
| Average value of f on [a, b] | 4 |
| Interval width (b − a) | 3 |
| Method | Adaptive Simpson's rule (numerical) |
Tips for using the Integration Calculator
- Check the tool with a known result: ∫₀³ (x² + 1) dx = [x³/3 + x]₀³ = 9 + 3 = 12.
- The function must be defined across the whole interval — 1/x from −1 to 1 has no finite integral.
- For the total (unsigned) area when the curve dips below the axis, integrate abs(f(x)).