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The Fundamental Theorem of Calculus explains why a definite integral can be calculated with an antiderivative: a running total of a continuous function changes at the rate given by that function, and the total accumulated change from a to b equals the antiderivative’s value at b minus its value at a.
What the Fundamental Theorem of Calculus says
The theorem has two closely related parts. The first connects accumulation to differentiation; the second gives a practical way to evaluate a definite integral. In the standard introductory versions, the function is assumed to be continuous on the interval in question.
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| Part | Given | Conclusion | In plain language |
|---|---|---|---|
| Accumulation and differentiation | A continuous function f and A(x) = ∫ax f(t) dt | A′(x) = f(x) | The running total changes at the current rate. |
| Endpoint evaluation | An antiderivative F such that F′ = f | ∫ab f(x) dx = F(b) − F(a) | Net accumulation is the change in an antiderivative. |
These statements are presented in introductory calculus materials from the University of Nebraska–Lincoln and the OpenStax Calculus Volume 1.
How a running total turns into a rate
Suppose f(t) describes a rate, such as how quickly water flows into a tank. Define A(x) as the total amount accumulated from a fixed starting point a up to a moving endpoint x:
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A(x) = ∫ax f(t) dt.
Move the endpoint a little, from x to x + Δx. The added total is the accumulation over that short slice. If f is continuous, then over a sufficiently small slice its value is close to f(x), so the added amount is approximately f(x)Δx. Dividing the change in the total by Δx and taking the limit gives A′(x) = f(x).
In other words, differentiation asks how fast the accumulated total is changing at one point; the answer is the function’s current value. The integration variable t is a placeholder, while x marks the moving endpoint. You could use another letter instead of t; the total would be unchanged. This notation helps distinguish what is being accumulated from where the accumulation ends. The University of Connecticut calculus text also explains the first-part relationship and the distinct roles of the two operations.
Why an antiderivative evaluates the integral
The second part makes the connection useful for calculation. If F′(x) = f(x), then the net accumulation of f from a to b is:
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∫ab f(x) dx = F(b) − F(a).
An antiderivative tracks a quantity whose rate of change is f. Its change from the start of the interval to the end is therefore the total net change contributed by that rate. Rather than adding infinitely many increasingly thin slices each time, you can find an antiderivative and subtract its endpoint values. The University of Nebraska–Lincoln’s second-part explanation and OpenStax’s section on the theorem show this evaluation rule in introductory examples.
Example: integrate x² from 0 to 1
An antiderivative of f(x) = x² is F(x) = x³/3, because differentiating x³/3 gives x². Apply the endpoint rule:
∫01 x² dx = F(1) − F(0) = 1³/3 − 0³/3 = 1/3.
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The subtraction matters: a definite integral over an interval is the change in an antiderivative across that interval, not just the antiderivative’s value at the right endpoint.
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A Riemann sum estimates an accumulated total by dividing an interval into pieces, multiplying a representative function value by each piece’s width, and adding the results. In the limit as the pieces become arbitrarily fine, that sum defines the definite integral in the standard introductory treatment.
The Fundamental Theorem does not replace that definition. Instead, it explains how the integral already defined by accumulation is linked to differentiation and antiderivatives. The first part says that differentiating the accumulation function recovers the original continuous function. The second says that an antiderivative can calculate the integral by endpoint subtraction. The University of Texas at Austin describes this as the connection between differentiation and integration; a learner discussion about Riemann sums and antiderivatives illustrates the kind of conceptual jump the theorem helps clarify.
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Why an integral can be negative
A definite integral measures signed accumulation, not automatically ordinary geometric area. Where the graph of f is above the horizontal axis, its contribution is positive; where it is below, its contribution is negative. Those negative contributions subtract from the net integral.
For example, if a rate is negative during part of a time interval, it represents a decrease in the accumulated quantity during that part. The definite integral records the resulting net change. If you need total geometric area between a graph and the axis, add the magnitudes of the portions below the axis rather than allowing them to cancel positive portions. The University of Kentucky’s introductory explanation discusses the theorem’s use with integrals; the distinction between signed integral and geometric area is essential when interpreting the result.
What “inverse operations” means here
It is tempting to say differentiation and integration are exact opposites, but that shorthand needs care. For a continuous f, differentiating its accumulation function returns f. In the other direction, integrating a derivative over an interval gives the net change in the original function: ∫ab F′(x) dx = F(b) − F(a).
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But knowing a derivative does not determine the original function’s starting value: functions that differ by a constant have the same derivative. That is why an antiderivative is not unique, even though the constant cancels when its endpoint values are subtracted. These standard statements assume the familiar introductory setting; more advanced versions handle broader conditions.
Where to practice
OpenStax Calculus Volume 1, Section 5.3 provides theorem explanations, worked examples, and exercises for practicing both the accumulation relationship and endpoint evaluation.
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