Table of Contents
The Fundamental Theorem of Calculus (often abbreviated FTC) is the key bridge between derivatives and definite integrals. In this chapter we will:
- State both parts of the theorem clearly.
- Explain what each part means.
- Show how they turn definite integrals into “antiderivative evaluations.”
- Give some basic examples and common pitfalls.
You should already be familiar with the ideas of definite integrals (as limits of Riemann sums) and antiderivatives from earlier chapters.
1. Two Parts of the Fundamental Theorem
The standard presentation breaks the Fundamental Theorem of Calculus into two related results, usually called Part 1 and Part 2.
Let $f$ be a real-valued function defined on an interval $[a,b]$.
1.1 Fundamental Theorem of Calculus, Part 1 (FTC1)
Statement (informal):
If $f$ is continuous on $[a,b]$, and we define a function $F$ by
$$
F(x) = \int_a^x f(t)\,dt ,
$$
then $F$ is differentiable on $(a,b)$ and
$$
F'(x) = f(x).
$$
So, “integrating $f$ from a fixed starting point to $x$” gives a new function $F$ whose derivative is the original function $f$.
Key points specific to FTC1:
- The integral has variable upper limit $x$ and fixed lower limit $a$.
- The function $F$ is sometimes called an accumulation function: it accumulates the area under $f$ from $a$ up to $x$.
- FTC1 tells us that every such accumulation function is an antiderivative of $f$ (as long as $f$ is continuous).
1.2 Fundamental Theorem of Calculus, Part 2 (FTC2)
Statement (informal):
If $f$ is continuous on $[a,b]$ and $F$ is any antiderivative of $f$ on $[a,b]$ (that is, $F'(x)=f(x)$ for all $x$ in $[a,b]$), then
$$
\int_a^b f(x)\,dx = F(b) - F(a).
$$
Key points specific to FTC2:
- The theorem gives a practical way to compute definite integrals:
- Find an antiderivative $F$ of $f$.
- Evaluate $F$ at the endpoints and subtract: “top minus bottom.”
- It doesn’t matter which antiderivative you pick; they all differ by a constant, and that constant cancels in $F(b)-F(a)$.
FTC2 is what turns definite integration from a limit-of-sums process into a straightforward evaluation problem, whenever $f$ has a known antiderivative.
2. Understanding Part 1: Accumulation and Differentiation
Consider a continuous function $f$ and define
$$
F(x) = \int_a^x f(t)\,dt.
$$
Intuitively, $F(x)$ measures the “total signed area” under the graph of $f$ from $t=a$ to $t=x$.
Now ask: How fast is this accumulated area changing when $x$ increases?
- When we increase $x$ a little bit from $x$ to $x+h$, we add a very thin strip of area under the curve.
- For small $h>0$, that extra area is approximately $f(x)\cdot h$ (height $\approx f(x)$, width $h$).
- So the rate of change of $F$ with respect to $x$ is about $f(x)$.
FTC1 says this intuitive picture is exact: the derivative $F'(x)$ is precisely $f(x)$.
2.1 Significance of FTC1
Specific insights from Part 1:
- It guarantees that if $f$ is continuous, then the accumulation function $F$ is differentiable and is an antiderivative.
- It shows that integration (as accumulation) and differentiation are inverse processes in a very concrete way.
- It gives a systematic way to construct an antiderivative of $f$: define $F(x)=\int_a^x f(t)\,dt$.
Note that FTC1 goes one way: “integrate then differentiate.” FTC2 completes the inverse relationship in a different direction: “differentiate (find an antiderivative) then integrate between bounds.”
3. Understanding Part 2: Evaluating Definite Integrals
Suppose $f$ is continuous and $F$ is an antiderivative of $f$. FTC2 says:
$$
\int_a^b f(x)\,dx = F(b) - F(a).
$$
This result connects:
- The definite integral $\int_a^b f(x)\,dx$ (a number, representing signed area)
to - Antiderivatives of $f$ (functions whose derivative is $f$).
3.1 Why the Endpoints Matter
The definite integral from $a$ to $b$ is the net accumulation from $a$ to $b$. If $F$ measures the accumulation from some reference point, then the total change in $F$ between $a$ and $b$ is
$$
F(b) - F(a).
$$
FTC2 tells us that this total change in an antiderivative $F$ is exactly the integral.
You do not need to know where $F$ is zero or how it is “anchored”; only the difference $F(b)-F(a)$ matters. This is why constants of integration never appear in definite integrals.
3.2 “Top Minus Bottom” Rule
In practice, FTC2 becomes the recipe:
- Find an antiderivative $F$ with $F'(x)=f(x)$.
- Compute $F(b)$.
- Compute $F(a)$.
- Subtract:
$$
\int_a^b f(x)\,dx = F(b) - F(a).
$$
You’ll often see a shorthand notation:
$$
\int_a^b f(x)\,dx = \left[ F(x) \right]_a^b = F(b)-F(a).
$$
The vertical bar with $a$ and $b$ is just notation reminding you to “evaluate at $b$ and $a$, then subtract.”
4. Examples Using the Fundamental Theorem
These examples focus on using FTC1 and FTC2; the skills to find antiderivatives themselves are developed in earlier chapters.
4.1 Using FTC2 to Evaluate a Simple Integral
Compute
$$
\int_0^3 (2x)\,dx.
$$
- Find an antiderivative: $F(x)=x^2$ since $F'(x)=2x$.
- Evaluate at endpoints:
- $F(3)=3^2=9$,
- $F(0)=0^2=0$.
- Subtract:
$$
\int_0^3 2x\,dx = F(3)-F(0) = 9-0 = 9.
$$
4.2 Integrating a Function with Negative Values
Compute
$$
\int_{-1}^2 (x-1)\,dx.
$$
- Antiderivative: $F(x)=\dfrac{x^2}{2}-x$.
- Evaluate:
- $F(2)=\dfrac{4}{2}-2=2-2=0$,
- $F(-1)=\dfrac{1}{2}-(-1)=\dfrac{1}{2}+1 = \dfrac{3}{2}$.
- Subtract:
$$
\int_{-1}^2 (x-1)\,dx = F(2)-F(-1) = 0 - \dfrac{3}{2} = -\dfrac{3}{2}.
$$
The negative value means the net signed area (area above the axis minus area below) is negative. FTC2 works with signed area, not just “unsigned” area.
4.3 Using FTC1 with a Variable Upper Limit
Define
$$
F(x)=\int_0^x \cos t\,dt.
$$
By FTC1, $F'(x) = \cos x$.
You can also compute $F(x)$ explicitly using FTC2:
- An antiderivative of $\cos t$ is $\sin t$.
- So
$$
F(x)=\int_0^x \cos t\,dt = \sin x - \sin 0 = \sin x - 0 = \sin x.
$$
Then $F'(x) = (\sin x)' = \cos x$, agreeing with FTC1.
Here we see both parts working together:
- FTC1 says: derivative of the accumulation function is the original integrand.
- FTC2 shows: that accumulation function is a particular antiderivative.
4.4 Nonzero Lower Limit in FTC1
FTC1 is often stated with $\int_a^x f(t)\,dt$. If instead you see
$$
G(x)=\int_c^x f(t)\,dt
$$
with some other constant $c$, the same conclusion holds:
$$
G'(x)=f(x).
$$
Changing the starting point only adds a constant to $G$, which does not affect the derivative.
5. Variants and Common Patterns
The simplest version of FTC1 uses an integral of the form
$$
F(x) = \int_a^x f(t)\,dt.
$$
In practice, you often encounter more complicated bounds or expressions. The following patterns are common applications derived from the Fundamental Theorem (together with the chain rule). The detailed derivations belong to differentiation techniques, so we only sketch the ideas here.
5.1 Variable Upper Limit with a Function Inside
Let
$$
H(x) = \int_a^{g(x)} f(t)\,dt,
$$
where $g$ is a differentiable function. Intuitively, as $x$ changes, the upper limit moves according to $g(x)$.
Under suitable conditions,
$$
H'(x) = f(g(x))\cdot g'(x).
$$
This result comes from combining FTC1 with the chain rule: differentiate the accumulation function evaluated at $g(x)$.
5.2 Both Limits Variable
If
$$
K(x) = \int_{u(x)}^{v(x)} f(t)\,dt,
$$
then, under appropriate conditions,
$$
K'(x) = f(v(x))\,v'(x) - f(u(x))\,u'(x).
$$
Intuitively:
- The upper limit contributes positively (“top” side).
- The lower limit contributes negatively (“bottom” side).
This is consistent with the idea that
$$
\int_{u(x)}^{v(x)} f(t)\,dt = \int_{a}^{v(x)} f(t)\,dt - \int_{a}^{u(x)} f(t)\,dt
$$
for a fixed $a$, and then applying FTC1 and the chain rule.
5.3 Reversing Limits
From the properties of the definite integral (and consistent with FTC2),
$$
\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx.
$$
This is especially useful when combined with FTC2:
- If you prefer to integrate from smaller to larger values, you can flip the limits and introduce a minus sign.
- For instance,
$$
\int_3^0 (2x)\,dx = -\int_0^3 (2x)\,dx.
$$
6. Conceptual Summary
The Fundamental Theorem of Calculus ties together the central ideas of calculus:
- Integration as accumulation:
The definite integral $\int_a^b f(x)\,dx$ represents net accumulation (signed area) of $f$ from $a$ to $b$. - Antiderivatives as “undoing” derivatives:
A function $F$ with $F'(x)=f(x)$ is an antiderivative of $f$. - FTC1:
If $F(x)=\int_a^x f(t)\,dt$, then $F'(x)=f(x)$.
Integration from a fixed starting point, then differentiation, returns the original continuous function. - FTC2:
If $F$ is any antiderivative of $f$, then
$$
\int_a^b f(x)\,dx = F(b)-F(a).
$$
Differentiation (finding an antiderivative) followed by evaluating at the endpoints computes the definite integral.
Together, these results show that differentiation and integration are deeply linked: they are inverse processes in a precise sense. This connection is what makes calculus such a powerful tool for analyzing change and accumulation.