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. Try the calculator itself right below, already loaded with the first example.
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This is the live calculator, already showing ∫(x^2, 0, 3) = 9. Edit the expression and press = to solve your own.
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What is a definite integral
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, not the total ink on the page. That single idea explains almost every result on this page, including the ones that look wrong at first.
How to use this definite integral calculator
- Make sure you're in COMP mode.
- If your function uses sin, cos or tan, tap the DEG pill until it reads Rad. The calculator opens in Degree mode by default, and that default is the single biggest reason a trig integral comes out wrong. More on this in Common mistakes below.
- Press the
∫template key, or type∫(directly. - 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)The calculation is numerical, not symbolic: the calculator estimates the area with an adaptive version of Simpson's rule rather than finding an antiderivative formula first. That is why it returns a number, not a step-by-step working, and why it can solve integrals that have no simple antiderivative at all.
Definite integral examples with solutions
Each example below shows the by hand method first, then a Try it link that opens the same expression in the live calculator so you can check the answer yourself.
∫(x^2, 0, 3) → 9. The antiderivative of x² is x³/3. Evaluate it at the limits: 3³/3 − 0³/3 = 27/3 − 0 = 9.
∫(sin(x), 0, π) → 2, in Rad mode. The antiderivative of sin(x) is −cos(x). Evaluate: −cos(π) − (−cos(0)) = 1 + 1 = 2. One arch of the sine curve has area exactly 2.
The calculator opens in Degree mode by default. Load this exact expression before switching to Rad and it computes sin of x degrees instead of x radians, returning about 0.08610697, a plausible-looking number, not an error. Tap the DEG pill, switch to Rad, and press = again to get the correct answer, 2. This is the single most common way to get a wrong definite integral.
∫(e^(x), 0, 1) → 1.718281828. The antiderivative of ex is ex itself. Evaluate: e¹ − e⁰ = e − 1.
∫(1/x, 1, 2) → 0.6931471806. The antiderivative of 1/x is ln(x). Evaluate: ln(2) − ln(1) = ln(2) − 0.
∫(x, -3, -1) → −4. The line y = x sits entirely below the x-axis for x between −3 and −1, so the signed area is negative. Antiderivative x²/2. Evaluate: (−1)²/2 − (−3)²/2 = 0.5 − 4.5 = −4. A negative result here is not a mistake, it is the curve living below the axis for the whole interval.
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) → 1/6. The calculator keeps this one as an exact fraction instead of a decimal, since 1/6 is a clean rational answer. Press S⇔D to see it as 0.1666666667 instead.
Common mistakes and how to avoid them
- Wrong angle mode is the single most common cause of a wrong integral. Trig functions inside
∫follow the calculator's DEG/Rad pill, the same as everywhere else, and the default is Degree.∫(sin(x), 0, π)is 2 in Rad mode and about 0.08610697 in Degree mode: a real, plausible looking number, not an error, so a wrong angle mode is easy to miss. Check the pill before you trust a trig integral. - Unexpected zero? Remember the integral is signed. An odd function like x³ integrated from −1 to 1 gives exactly
∫(x^3, -1, 1)= 0, because the negative area on [−1, 0] cancels the positive area on [0, 1] (try it →). If you wanted the total, unsigned area instead, split the integral at the root:∫(x^3, 0, 1)is the exact fraction 1/4 (try it →), and doubling that by hand gives 1/2 for the full interval. - Bracket bug:
1/x+1is (1/x)+1. For 1/(x+1) write the brackets explicitly. - Error or a huge number? The interval probably contains a vertical asymptote, such as 1/x integrated across 0. The calculator cannot integrate through a point where the function is undefined.
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?
Open the calculator in COMP mode, press the ∫ template key, and enter ∫(f(x), lower, upper). For example, ∫(x^2, 0, 3) evaluates to 9. Press = to get the result.
Why does my definite integral return a negative number?
A definite integral measures signed area. Regions where the function is below the x-axis count as negative, so the net result can be negative or zero even when there is area on both sides.
Do I need radians for trigonometric integrals?
Yes. Trig functions inside ∫ follow the calculator's angle mode pill, the same as everywhere else on the calculator. Set it to Rad before integrating sin, cos or tan. In Degree mode, the default, the result is a different number, not an error, which is why the mistake is easy to miss.
Can it find the area between two curves?
Yes, integrate the difference of the two functions over the interval, ∫(top(x) − bottom(x), a, b). The result is the enclosed area when top(x) ≥ bottom(x) on [a, b].
What is the difference between a definite integral and an indefinite integral?
A definite integral has two limits and returns one number: the signed area under the curve between them. An indefinite integral has no limits and returns a family of antiderivative functions instead of a number. This calculator solves definite integrals only, and it works numerically, so the answer is always a number, not a formula.
How do you calculate a definite integral by hand?
Find the antiderivative F(x) of the function, then compute F(upper) minus F(lower). For ∫(x^2, 0, 3), the antiderivative is x³/3, so the answer is 3³/3 minus 0³/3, which is 9. The calculator reaches the same number a different way: it estimates the area numerically instead of finding F(x) first, which is why it still works for functions with no simple antiderivative.
Can I use this as an area under a curve calculator?
Yes. When the curve stays on or above the x-axis over your interval, the definite integral is the area under it. Enter ∫(f(x), lower, upper) the same way as any other definite integral. If the curve dips below the axis, remember the result is signed area, so for the true, unsigned area you would split the interval at each root and add the absolute values of each piece.
Does the calculator show the steps, or just the final answer?
Just the final numeric answer. It solves the integral by numerical approximation (adaptive Simpson's rule), not by finding the antiderivative step by step, so there is no symbolic working to display. The worked examples on this page show the by hand steps for the same problems, so you can check its answers against the method a textbook would use.
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Open the calculator, press ∫, and reproduce any example above in seconds.
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