Definite Integral Calculator

The definite integral is the workhorse of applied calculus: it turns a rate into an accumulated total, a curve into an area, a velocity record into a distance. Evaluating one by hand means finding an antiderivative and then carefully subtracting its values at two endpoints — a short procedure, but one where bracket errors and sign slips are remarkably common. This calculator evaluates definite integrals directly, which makes it a fast check on exam work and a practical tool for anyone who needs the number more than the derivation.

Quick Answer

By the Fundamental Theorem of Calculus, ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f. The constant of integration cancels out, which is why a definite integral produces a single number rather than a family of functions.

What the Number Actually Represents

Geometrically, ∫ₐᵇ f(x) dx is the signed area between the curve and the x-axis from x = a to x = b. Regions above the axis count positive; regions below count negative. That word "signed" matters: if a curve dips below the axis, the integral subtracts that part instead of adding it, so the result is a net total, not a measurement of total enclosed area. If you need the physical area regardless of sign, the interval must be split at each root of f.

The same mathematics wears many costumes. Integrate a velocity and you get displacement. Integrate a flow rate and you get the volume delivered. Integrate a marginal cost and you get the total cost of a production run. In every case the structure is identical: an antiderivative evaluated at the top limit, minus the same antiderivative at the bottom limit. The order is not negotiable — F(b) − F(a), top minus bottom. Swapping the limits flips the sign of the answer, a property that is sometimes useful and often the source of an avoidable mistake.

How to Use the Definite Integral Calculator

1. Enter the integrand f(x) exactly as written, with brackets around grouped terms.
2. Enter the lower limit a and the upper limit b, checking that a really is the smaller value (if it is not, expect a sign flip).
3. Confirm the variable of integration matches the variable in your function.
4. Run the calculation and note whether the answer is exact (a fraction or expression) or decimal.
5. If the curve crosses the x-axis inside the interval and you need total area rather than net area, split the integral at the crossing points and evaluate the pieces separately.
6. Sanity-check the size: a positive function on a short interval should give a modest positive number; a wildly large answer usually means a misplaced bracket.

A Worked Example: ∫₀² (3x² + 2x) dx

An antiderivative of 3x² + 2x is F(x) = x³ + x². Evaluate at the top limit: F(2) = 8 + 4 = 12. Evaluate at the bottom: F(0) = 0. Subtract: 12 − 0 = 12. The example is deliberately clean, but the rhythm is the same for every definite integral — antiderivative, top value, bottom value, subtract. Try a slightly harder mental check with the same method: ∫₁³ 2x dx has antiderivative x², giving 9 − 1 = 8, which matches the area of the trapezoid under the line from x = 1 to x = 3 (width 2, heights 2 and 6, average height 4, area 8). When an integral and a simple geometric area agree, both are almost certainly right.

Where Definite Integral Calculations Break Down

The most frequent error is forgetting the subtraction entirely and reporting F(b) as the answer. Close behind is evaluating the antiderivative at the limits in the wrong order. A subtler trap is applying the method across a discontinuity: if the function blows up or is undefined inside [a, b], the ordinary procedure can produce a plausible-looking number for an integral that is actually improper and may diverge. Watch units as well — integrating a rate measured per hour over an interval measured in minutes silently scales the answer by 60. For the antiderivative step itself, the companion Indefinite Integral Calculator shows the F(x) you need, and for the geometric reading of the result, the Area Under the Curve Calculator focuses on the enclosed region.

From Accumulation to Average Value

One application of the definite integral deserves its own mention because it appears in every course and many workplaces: the average value of a function over an interval. The definition is the integral you already know, divided by the interval's length: average = [1/(b − a)] · ∫ₐᵇ f(x) dx. It answers questions like "what was the average temperature today?" from a continuous temperature record, or "what constant rate would have delivered the same total?" Using the worked example above, the average value of 3x² + 2x on [0, 2] is 12 / 2 = 6 — a constant function at height 6 over the same interval would enclose the same area and accumulate the same total. Geometrically, it is the height of the rectangle on [a, b] whose area equals the integral. Notice that the average value is not the average of the endpoint values — for this function that would be (0 + 16)/2 = 8, which overstates it, because the curve spends more of the interval at low values than at high ones. The integral weights every point of the interval equally; the endpoints-only shortcut weights two points and ignores the rest.

Units provide a final, underused check on any definite integral. The integral of a rate carries the units of the quantity it accumulates: integrate litres per minute over minutes and the answer is in litres; integrate metres per second over seconds and it is in metres. If your computed number's units do not match the quantity the problem asked for, the setup — not the arithmetic — is wrong, most often because the limits were entered in different units from the rate. Checking units takes seconds and catches a class of error that re-doing the algebra never will, because the algebra was never the problem.

Taking the Accumulated Total Further

Once an accumulation is trustworthy, it becomes an input to bigger questions — average value of a function, work done by a variable force, probability from a density. Verify your integral here first, then explore the broader Integral Calculator for techniques such as substitution and parts when the antiderivative is not obvious.