A definite integral calculator online turns the area-under-a-curve problem into a single keystroke sequence. This guide shows exactly how to enter one on our free scientific calculator, with examples you can reproduce and a clear explanation of why some answers come out negative or zero.
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What a definite integral measures
A definite integral ∫ab f(x) dx is the signed area between the curve y = f(x) and the x-axis, from x = a to x = b. "Signed" is the key word: area above the axis is positive, area below is negative, and the integral reports the net total. That single idea explains almost every surprising result you'll see.
Entering one, step by step
- Make sure you're in COMP mode.
- If your function uses trig, set the angle mode to Rad.
- Press the
∫template key. - Type the integrand in terms of
x. - Enter the lower limit, then the upper limit, separated by commas.
- Press
=.
The general form is:
∫(f(x), lower, upper)Five worked examples
∫(x^2, 0, 3) → 9. Antiderivative x³/3 from 0 to 3 = 27/3 = 9. ✔
∫(sin(x), 0, π) → 2. One arch of the sine curve has area exactly 2.
∫(1/x, 1, 2) → 0.6931… = ln 2.
∫(e^(x), 0, 1) → 1.7182… = e − 1.
∫(x^3, -1, 1) → 0. x³ is odd, so the negative area on [−1, 0] cancels the positive area on [0, 1].
If you actually wanted the total (unsigned) area, integrate the absolute value or split the integral at the root: ∫(x^3, 0, 1) doubles to 0.5.
Area between two curves
To find the region enclosed between an upper curve and a lower curve on [a, b], integrate their difference:
∫(top(x) − bottom(x), a, b)Area between y = x and y = x² on [0, 1]: ∫(x − x^2, 0, 1) → 0.1666… = 1/6.
When the answer looks wrong
- Unexpected sign or zero? Remember the integral is signed — see example 5.
- Trig nonsense? You're probably in Deg mode. Switch to Rad.
- Bracket bug:
1/x+1is (1/x)+1. For 1/(x+1) write the brackets explicitly. - Error or huge number? The interval may contain a vertical asymptote (e.g. 1/x across 0).
These and more are covered in common integration mistakes students make. For the wider picture of how the tool's calculus engine works, see the pillar guide on using an online integration & derivative calculator, and the companion on numerical differentiation.
Frequently asked questions
How do I compute a definite integral online?
In COMP mode press the ∫ key and enter ∫(f(x), lower, upper) — e.g. ∫(x^2, 0, 3) = 9 — then press =.
Why does my integral return a negative number?
A definite integral measures signed area; regions below the x-axis subtract, so the net can be negative or zero.
Do I need radians for trig integrals?
Yes — set the angle mode to Rad before integrating sin, cos or tan.
Can it find the area between two curves?
Yes — integrate top(x) − bottom(x) over the interval.
Compute your integral now
Open the calculator, press ∫, and reproduce any example above in seconds.
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