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The Fundamental Theorem of Calculus explains why differentiation and integration work together: a running total changes at the rate being accumulated, and a definite integral can be calculated by subtracting an antiderivative’s endpoint values.
What the theorem says
Imagine a function describing a rate, such as how quickly something is changing. Add up that rate from a fixed starting point to a point that moves along the number line. The result is an accumulated total.
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If f is continuous and we define A(x) = ∫ax f(t) dt, then the first part of the theorem says:
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In plain language, the running total changes at the current rate. The variable t is just a name for the input being accumulated; using it keeps that input distinct from the moving endpoint x.
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How an antiderivative calculates a definite integral
The second part gives a practical way to evaluate a definite integral. If F′(x) = f(x) on the interval, then:
∫ab f(x) dx = F(b) − F(a)
Rather than adding up infinitely many increasingly thin slices each time, find an antiderivative F, evaluate it at the interval’s end and start, and subtract. The integral is the net accumulation across the interval, represented by the change in F.
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Example: integrate x² from 0 to 1
An antiderivative of x² is F(x) = x³/3, because its derivative is x². Therefore:
∫01 x² dx = F(1) − F(0) = 1/3 − 0 = 1/3.
The endpoint subtraction gives the integral directly. It does not mean that the integral is defined by antiderivatives: in the usual introductory treatment, the definite integral is defined through sums, and the theorem connects that accumulated total to antiderivatives.
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How the two parts fit together
The theorem links local change with total accumulation. Its two forms start from different information and answer different questions:
| Form | Given | Find | Interpretation |
|---|---|---|---|
| Accumulation derivative | 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 with F′ = f | ∫ab f(x) dx = F(b) − F(a) | Net accumulation equals the change in an antiderivative. |
These results are related, but they are not a license to say that differentiation and integration are exact inverses in every sense. Differentiating an accumulation function recovers the integrand under the theorem’s continuity condition. Recovering a function from its derivative determines it only up to an additive constant.
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Why an integral can be negative
A definite integral measures signed accumulation. Where the graph of f is above the horizontal axis, it contributes positively; where it is below, it contributes negatively. So the integral is not always the same as the ordinary geometric area between the graph and the axis.
If you want total geometric area and the function crosses the axis, treat the regions above and below separately and count each region’s area as positive. The signed integral instead keeps the below-axis contribution negative, allowing positive and negative parts to cancel.
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From Riemann sums to the antiderivative shortcut
A Riemann sum estimates accumulation by splitting an interval into thin pieces, multiplying each piece’s width by a function value, and adding the products. As the pieces become finer, the sums approach the definite integral. The Fundamental Theorem then tells you that this same accumulated quantity can be found from an antiderivative’s endpoint values.
That is the bridge: sums describe what the integral accumulates; the theorem explains why an antiderivative makes the total easier to calculate.
Where to study the theorem further
OpenStax Calculus Volume 1, Section 5.3 presents the theorem with examples and exercises.
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