Fundamental theorem of calculus
Evaluate signed accumulated change using any antiderivative.
Fundamental theorem of calculus is one of 7 integration formulas in the calculus section of this library, and it is used at ap · university level.
Why fundamental theorem of calculus works
An integral adds up infinitely many thin slices, and each slice is the small amount of change the antiderivative makes across that step. Adding every bit of change from one end to the other simply tells you how far the antiderivative moved in total, which is its ending value minus its starting value.
What each symbol means
$F$ is an antiderivative and $a,b$ are bounds.
Fundamental theorem of calculus: when it holds
$f$ must satisfy the theorem’s integrability conditions, commonly continuity on $[a,b]$.
When it stops applying
The function has to be integrable across the whole closed interval, which normally means continuous there. Applied blindly across a vertical asymptote it produces nonsense: the integral of 1 over x squared from −1 to 1 appears to be −2, even though the integrand is positive everywhere it is defined.
Fundamental theorem of calculus: a worked example
$\int_0^2 3x^2dx=[x^3]_0^2=8$.
The mistake to avoid
What people do: Subtracting in the wrong order, taking the value at the lower limit minus the value at the upper limit.
Why it goes wrong: The order carries the direction of travel, so the answer arrives with its sign flipped and any area is reported as negative.
Do this instead: Always take the top value first and subtract the bottom value from it.
Fundamental theorem of calculus: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $F$ is an antiderivative and $a,b$ are bounds.
- Check the conditions before substituting. $f$ must satisfy the theorem’s integrability conditions, commonly continuity on $[a,b]$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Integration
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where fundamental theorem of calculus comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about fundamental theorem of calculus
Does it matter which antiderivative I use?
No. Any two antiderivatives of the same function differ by a fixed constant, and that constant is removed by the subtraction, so every choice gives one identical number.
Why can a definite integral come out negative?
Because area below the horizontal axis is counted as negative. The theorem reports signed accumulation, so for a total geometric area you split the interval where the function crosses zero.
How does this relate to the other half of the theorem?
The other half says that differentiating an accumulation gives the integrand back. This half says an antiderivative evaluates the accumulation, so they are one statement read in two directions.
Do I write a constant of integration here?
No. It would cancel in the subtraction anyway, which is exactly why a definite integral produces a single number rather than a whole family of answers.