Integral Calculator guide
Integral Calculator: Definite Integrals and Area Under a Curve
Use our integral calculator to work out definite integrals and the area under a curve between two limits, with examples and the difference between signed and total area.
By the AllCalculator editorial teamUpdated 15 October 20262 min read
Integration adds up lots of tiny pieces to find a total, such as the area under a curve or the distance travelled from a speed. Our Integral Calculator works out definite integrals between any two limits.
How to use the calculator (Integral Calculator)
- Type your function of x, for example x^2 + 1.
- Enter the lower and upper limits. You can use pi and e, such as pi/2.
- Read the definite integral, the total area and the average value.
Signed area and total area
The definite integral counts area above the x-axis as positive and area below as negative. So the integral of sin(x) from 0 to 2π is 0, because the two halves cancel out.
The calculator also shows the total area, which counts every part as positive. For sin(x) from 0 to 2π, that's 4.
A worked example
With the calculator's default settings, the integral of x² + 1 from 0 to 3:
- The antiderivative is x³/3 + x
- At 3: 9 + 3 = 12. At 0: 0
- So the integral is 12
The average value of the function over the interval is 12 ÷ 3 = 4.
How the calculator works it out
It uses adaptive Simpson's rule, a numerical method that fits curves through points and keeps splitting the interval until the answer is accurate to many decimal places. It works for functions you couldn't integrate by hand.
Common antiderivatives
| Function | Integral |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ / (n + 1) |
| 1/x | ln(x), for x > 0 |
| eˣ | eˣ |
| sin x | −cos x |
| cos x | sin x |
Plot the curve first with our Graphing Calculator, or find gradients with our Derivative Calculator.
Frequently asked questions
Can it do indefinite integrals?
It works out definite integrals, which give a number. For indefinite integrals, use the table above and add + C.
Why is my integral negative?
More of the curve is below the x-axis than above it between your limits. Check the total area figure for the area regardless of sign.
What if the function is undefined inside the range?
The integral may not exist, for example 1/x from −1 to 1. The calculator will tell you if it can't converge.
Integration is the other half of calculus. Use the Integral Calculator to check your working.
Sources and further reading
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.