Table of Contents
Overview
Integral calculus studies accumulation. Where differential calculus focuses on instantaneous change (derivatives), integral calculus focuses on “total effect” over an interval: total distance traveled from a speed function, total area under a curve, total mass from a density, and so on.
This chapter gives the big picture of integrals and how the main ideas fit together. Later chapters in this part of the course (Indefinite Integrals, Definite Integrals, Applications of Integrals) will handle the details.
The Two Main Types of Integrals
Integral calculus revolves around two related but distinct concepts:
- Indefinite integrals
These represent families of antiderivatives. When you see
$$
\int f(x)\,dx,
$$
without limits on the integral sign, you are looking at an indefinite integral. The result is a function (plus a constant), not a single number. - Definite integrals
These represent accumulated quantities over an interval. When you see
$$
\int_a^b f(x)\,dx,
$$
with numbers $a$ and $b$ as limits, you are looking at a definite integral. The result is a number, often interpreted as “signed area under the curve” or “total accumulated change” between $x=a$ and $x=b$.
The later sections in this part of the course are organized around this distinction: first indefinite integrals and antiderivatives, then definite integrals and the Fundamental Theorem of Calculus, and finally applications.
Antiderivatives and Indefinite Integrals
In differential calculus, you learned how to go from a function to its derivative. Integral calculus often asks you to reverse this process.
If a function $F(x)$ has derivative $F'(x) = f(x)$, then $F$ is called an antiderivative of $f$. There are infinitely many antiderivatives: if $F$ is one, then $F+C$ is also an antiderivative of $f$ for any constant $C$, because
$$
\frac{d}{dx}[F(x)+C] = F'(x) = f(x).
$$
The indefinite integral is a way to represent all antiderivatives of a function at once. We write
$$
\int f(x)\,dx = F(x) + C,
$$
where $F'(x) = f(x)$. The symbol $\int$ (an elongated “S”) indicates integration, and the $dx$ indicates the variable of integration.
In the dedicated chapter on Indefinite Integrals, you will learn:
- How to interpret $\int f(x)\,dx$ as “find an antiderivative of $f(x)$.”
- Basic antiderivatives of common functions.
- Systematic methods for finding antiderivatives in more complex situations.
Definite Integrals and Accumulation
Where indefinite integrals produce functions, definite integrals produce numbers. Conceptually, a definite integral measures the total accumulation of a quantity as the variable moves from $x=a$ to $x=b$.
For a continuous function $f$ on the interval $[a,b]$, the definite integral
$$
\int_a^b f(x)\,dx
$$
can be thought of as the net area between the graph of $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$:
- Regions where $f(x) > 0$ contribute positive area.
- Regions where $f(x) < 0$ contribute negative area.
A key idea underlying the definite integral is to approximate this area (or accumulated quantity) by adding up the areas of many thin rectangles. As the rectangles get thinner and more numerous, the approximation improves. The formal process of taking the limit of these sums is developed through Riemann sums in the later chapter on Definite Integrals.
Definite integrals are used to compute:
- Areas under curves.
- Total distance traveled from a velocity function.
- Total mass from a variable density.
- Accumulated profit, cost, or other quantities that vary continuously.
The Fundamental Theorem of Calculus (Conceptual View)
The central bridge between derivatives and integrals is the Fundamental Theorem of Calculus (FTC). While the formal statement and proofs belong in the Definite Integrals chapter, it is important here to understand its conceptual role.
In broad terms, the FTC tells you two powerful facts:
- Accumulation function has derivative equal to the integrand.
If you accumulate the area under a curve $f(x)$ from a fixed starting point up to a variable endpoint $x$, the rate at which this accumulated area changes is just $f(x)$ itself. - Definite integral can be computed using antiderivatives.
If $F$ is any antiderivative of $f$ (so $F'(x) = f(x)$), then the definite integral from $a$ to $b$ is given by
$$
\int_a^b f(x)\,dx = F(b) - F(a).
$$
In practice, this means: to compute a definite integral, you do not need to go back to sums of rectangles; instead, you find an antiderivative (an indefinite integral) and evaluate it at the endpoints.
Thus, indefinite integrals (antiderivatives) and definite integrals (accumulation) are mathematically tied together by the Fundamental Theorem of Calculus.
Geometric vs. Physical Interpretations
Integral calculus is rich in interpretation. Two of the most important viewpoints are:
- Geometric interpretation:
The definite integral $\int_a^b f(x)\,dx$ measures the net signed area between the graph of $f$ and the $x$-axis on $[a,b]$. - Physical interpretation:
Many physical quantities can be interpreted as the integral of a rate: - If $v(t)$ is velocity, then $\int_a^b v(t)\,dt$ is the change in position from time $t=a$ to $t=b$.
- If $r(t)$ is a rate of flow (such as water into a tank), then $\int_a^b r(t)\,dt$ is the total amount that has flowed in during $[a,b]$.
- If $p(x)$ is a probability density function, then $\int_a^b p(x)\,dx$ is the probability that a random variable lies between $a$ and $b$.
The Applications of Integrals chapter will explore these and other situations, but the unifying idea is always the same: a rate or density integrated over an interval gives a total amount.
Notation and Language of Integration
Across all of the integral calculus topics, several pieces of notation and terminology are standard:
- The symbol $\int$ indicates an integral. You may think of it as a stylized “S” standing for “sum,” because integral calculus formalizes the idea of adding up infinitely many tiny contributions.
- In an expression like
$$
\int_a^b f(x)\,dx,
$$
the quantity $f(x)$ is called the integrand, the numbers $a$ and $b$ are the limits of integration, and $dx$ indicates the variable of integration. - If you change variables, you must also change the differential. For example:
$$
\int g(t)\,dt
$$
uses $t$ instead of $x$ and has $dt$ instead of $dx$. - A definite integral, with limits, evaluates to a number.
An indefinite integral, without limits, evaluates to a family of functions, usually written with a $+C$ to represent the constant of integration.
Later chapters will also introduce and use specific notation for numerical approximations (like $\Delta x$ for small interval widths in Riemann sums) and for application-specific quantities (like $v(t)$ for velocity).
Conceptual Skill Set in Integral Calculus
By the end of the Integral Calculus part of the course (including its subsections), you should be comfortable with the following conceptual abilities:
- Recognizing when a problem is asking for:
- An antiderivative (indefinite integral), or
- A total accumulated quantity (definite integral).
- Interpreting $\int_a^b f(x)\,dx$ as a limit of sums and as an area or accumulation.
- Using antiderivatives to compute definite integrals via the Fundamental Theorem of Calculus.
- Translating a real-world rate or density into an integrand and appropriate limits of integration.
- Understanding at a conceptual level when the sign of the integrand matters (net accumulation versus total magnitude, for example in signed area versus total distance).
The subsequent chapters—Indefinite Integrals, Definite Integrals, and Applications of Integrals—will provide the techniques, details, and practice needed to carry out these ideas in concrete situations.