Fundamental Theorem of Calculus

students, this lesson connects two big ideas in calculus and real analysis: differentiation and integration. 🌟 The surprise is that they are not separate topics at all. The Fundamental Theorem of Calculus shows how integration can be undone by differentiation, and how a derivative can be recovered from an accumulated area function.

What you will learn

What the Fundamental Theorem of Calculus says in precise mathematical language.
How to use it to compute definite integrals more efficiently.
Why the theorem matters in Real Analysis and Riemann Integration II.
How to interpret an integral as accumulation and a derivative as a rate of change.
How the theorem fits with integrability criteria and the properties of the integral.

Imagine a water tank being filled over time 🚰. The function $f(t)$ gives the filling rate at time $t$. The total amount of water added from time $a$ to time $b$ is the integral $\int_a^b f(t)\,dt$. The Fundamental Theorem of Calculus explains how this total accumulation changes as the endpoint moves, and how to recover the original rate function from an antiderivative.

The two parts of the theorem

The Fundamental Theorem of Calculus is usually split into two parts.

Part 1: Differentiating an accumulation function

Suppose $f$ is integrable on $[a,b]$, and define a new function by

$$F(x)=\int_a^x f(t)\,dt.$$

This function $F$ records the total signed area from $a$ up to $x$. The theorem says that if $f$ is continuous at a point $x$, then $F$ is differentiable at that point and

$$F'(x)=f(x).$$

This is powerful because it says the derivative of the accumulated area function gives back the original integrand. In words: if $f$ is the rate, then $F$ is the total, and the slope of the total at a point is the current rate.

A simple example is $f(t)=t^2$ on $[0,3]$. Then

$$F(x)=\int_0^x t^2\,dt.$$

Using the power rule for antiderivatives, we get

$$F(x)=\frac{x^3}{3},$$

$$F'(x)=x^2=f(x).$$

This matches the theorem exactly.

Part 2: Evaluating definite integrals with antiderivatives

If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ on that interval, meaning

$$F'(x)=f(x),$$

then

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

This is the most famous computational use of the theorem. Instead of estimating area with rectangles, we can often find an antiderivative and subtract values.

For example, if we want

$$\int_1^4 3x^2\,dx,$$

we look for an antiderivative of $3x^2$, which is $x^3$. Then

$$\int_1^4 3x^2\,dx=4^3-1^3=64-1=63.$$

This is much faster than computing the area directly from the definition of the Riemann integral.

Why the theorem is true in Real Analysis

In Real Analysis, the theorem is not just a formula to memorize. It rests on careful ideas about limits, continuity, and Riemann integrability.

The function $F(x)=\int_a^x f(t)\,dt$ is built from a Riemann integral. To study its derivative at a point $x$, we look at the difference quotient

$$\frac{F(x+h)-F(x)}{h}.$$

Substituting the definition of $F$ gives

$$\frac{1}{h}\int_x^{x+h} f(t)\,dt.$$

If $f$ is continuous at $x$, then on a very small interval near $x$, the values of $f(t)$ stay close to $f(x)$. So the average value of $f$ over that tiny interval approaches $f(x)$ as $h\to 0$. That is why

$$\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}=f(x).$$

This reasoning uses the idea that continuous functions behave predictably on small intervals. It also shows why the theorem may fail at a point where $f$ is badly discontinuous. The theorem does not require $f$ to be differentiable; continuity at the relevant point is enough for Part 1.

A key takeaway for students is that the theorem bridges local behavior and global accumulation. The derivative is local, while the integral is global. The FTC proves these two viewpoints are deeply connected.

Using the theorem to compute integrals

A central skill in Riemann Integration II is choosing the right tool for a definite integral. The FTC often turns a difficult integral into a simple substitution of endpoints.

Example 1: Polynomial integrals

Compute

$$\int_2^5 (4x^3-2x)\,dx.$$

An antiderivative is

$$F(x)=x^4-x^2.$$

$$\int_2^5 (4x^3-2x)\,dx=F(5)-F(2)=(625-25)-(16-4)=600-12=588.$$

Example 2: Trigonometric integrals

Compute

$$\int_0^{\pi} \sin x\,dx.$$

An antiderivative of $\sin x$ is $-\cos x$, so

$$\int_0^{\pi} \sin x\,dx=(-\cos \pi)-(-\cos 0)=1-(-1)=2.$$

This matches the geometric interpretation that the area under $\sin x$ from $0$ to $\pi$ is positive and equals $2$.

Example 3: A function with no elementary antiderivative

Some integrals cannot be expressed using elementary functions. For instance,

$$\int_0^1 e^{-x^2}\,dx$$

does not have an elementary antiderivative. The FTC still tells us the integral defines an accumulation function, and that function can be studied even if it cannot be written in a simple closed form. This is an important Real Analysis lesson: the integral is meaningful even when no simple formula exists.

How the theorem fits with properties of the integral

The Fundamental Theorem of Calculus works together with other properties of the Riemann integral.

If $f$ and $g$ are integrable on $[a,b]$, then so is $f+g$, and

$$\int_a^b (f(x)+g(x))\,dx=\int_a^b f(x)\,dx+\int_a^b g(x)\,dx.$$

Also, if $c$ is a constant,

$$\int_a^b c f(x)\,dx=c\int_a^b f(x)\,dx.$$

These linearity properties help build antiderivatives from simpler pieces.

The theorem also interacts with the fact that continuous functions on a closed interval are Riemann integrable. Since continuous functions are integrable, Part 1 can be applied to define accumulation functions such as

$$F(x)=\int_a^x f(t)\,dt.$$

Then Part 2 says that if $F'(x)=f(x)$, the integral from $a$ to $b$ can be computed by endpoint subtraction.

Another important idea is that the integral depends only on the values of the function almost everywhere for many advanced results, but in the Riemann setting, the exact pointwise behavior still matters for integrability criteria. For FTC, continuity is especially useful because it guarantees the accumulation function behaves smoothly enough to differentiate.

Intuition: area, accumulation, and slope

Think of a delivery app tracking packages over time 📦. If $f(t)$ counts the rate at which packages are delivered per hour, then

$$F(x)=\int_a^x f(t)\,dt$$

counts the total packages delivered from time $a$ to time $x$. If the delivery rate at time $x$ is high, then the total count increases quickly, so $F'(x)$ is large. If the delivery rate is low, $F'(x)$ is small.

This intuition makes the theorem feel natural:

the integral collects small contributions into a total,
the derivative measures how fast that total is changing,
and the FTC says the change rate of the total is the original rate.

In a graph, if $f(x)$ is positive, the accumulation function $F(x)$ increases. If $f(x)$ is negative, $F(x)$ decreases. If $f(x)=0$, then $F(x)$ is flat at that moment. The sign and size of $f(x)$ control the slope of $F$.

Common misunderstandings to avoid

A few details matter in Real Analysis.

First, the FTC does not say every integrable function is differentiable. The integrand $f$ may be merely integrable, while the accumulation function $F(x)=\int_a^x f(t)\,dt$ is the one that becomes differentiable at points where $f$ is continuous.

Second, not every antiderivative comes from a simple algebraic formula. Some functions have antiderivatives, but writing them explicitly may be impossible with elementary functions.

Third, the theorem uses definite integrals with fixed limits. The result

$$\int_a^b f(x)\,dx=F(b)-F(a)$$

works because the limits are endpoints of an interval, not because integration is a purely symbolic process.

Finally, the theorem is about a rigorous limit process. It is not just a shortcut. The proof depends on the definition of the Riemann integral and the behavior of continuous functions on small intervals.

Conclusion

The Fundamental Theorem of Calculus is one of the central results of Real Analysis because it unifies two major ideas: accumulation and change. Part 1 says that the derivative of an integral accumulation function recovers the original integrand at points of continuity. Part 2 says that if a function has an antiderivative, then its definite integral is found by subtracting endpoint values. Together, these facts make integration both conceptually meaningful and computationally powerful. For students, this theorem is a key bridge in Riemann Integration II, connecting the properties of the integral, integrability, and the deeper relationship between differentiation and integration. ✨

Study Notes

The function $F(x)=\int_a^x f(t)\,dt$ is an accumulation function.
If $f$ is continuous at $x$, then $F'(x)=f(x)$.
If $F'(x)=f(x)$ on $[a,b]$, then

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

The FTC links local slope information with global area or accumulation.
Continuous functions on closed intervals are Riemann integrable, so the theorem applies often in practice.
Linearity of the integral helps simplify integrals before applying the FTC.
Some functions have no elementary antiderivative, but the integral still exists and is meaningful.
The theorem is a central result in Riemann Integration II and a foundation for more advanced analysis.