KAHIBARO
Table of Contents
Accumulation and Area
An integral is a mathematical tool for adding up many tiny contributions. In physics, this is useful whenever a quantity builds up continuously over space, time, or some other variable. If a quantity changes from point to point, simple multiplication is often not enough. An integral lets us combine all the small pieces into one total.
A common first idea is area under a curve. If a graph of a function $f(x)$ is drawn against $x$, the integral of $f(x)$ over an interval gives the signed area between the curve and the axis. "Signed" means areas above the axis count as positive and areas below count as negative.
If we divide an interval from $a$ to $b$ into many small pieces of width $\Delta x$, then on each piece the contribution is approximately $f(x)\Delta x$. Adding all pieces gives an approximation to the total. In the limit of infinitely small pieces, this becomes the definite integral:
$$
\int_a^b f(x)\,dx
$$
Here, $f(x)$ is the function being added, $a$ and $b$ are the limits, and $dx$ represents a very small change in $x$.
The definite integral
$$
\int_a^b f(x)\,dx
$$
represents the total accumulation of $f(x)$ from $x=a$ to $x=b$.
It is not just an algebraic symbol. It means the limit of a sum of many tiny contributions.
Definite and Indefinite Integrals
There are two main kinds of integrals. The indefinite integral gives a family of functions, while the definite integral gives a number.
The indefinite integral is written as
$$
\int f(x)\,dx
$$
and means, find a function whose derivative is $f(x)$. This function is called an antiderivative. Since the derivative of a constant is zero, the answer always includes an arbitrary constant $C$:
$$
\int f(x)\,dx = F(x) + C
$$
where
$$
\frac{dF}{dx} = f(x)
$$
The definite integral is written with limits:
$$
\int_a^b f(x)\,dx
$$
and gives the accumulated value between two points.
These two ideas are connected by the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links integration and differentiation. If $F(x)$ is an antiderivative of $f(x)$, then
$$
\int_a^b f(x)\,dx = F(b) - F(a)
$$
This means that to evaluate many definite integrals, we first find an antiderivative and then subtract its values at the endpoints.
If
$$
\frac{dF}{dx} = f(x),
$$
then
$$
\int_a^b f(x)\,dx = F(b) - F(a)
$$
This is one of the most important formulas in calculus and physics.
For example, if $f(x) = x^2$, then an antiderivative is
$$
F(x) = \frac{x^3}{3}
$$
so
$$
\int_0^2 x^2\,dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3}
$$
Basic Integration Rules
Many integrals are found by reversing derivative rules. Some common examples are shown below.
| Function | Indefinite integral | ||
|---|---|---|---|
| $x^n$, for $n \neq -1$ | $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ | ||
| $\frac{1}{x}$ | $\int \frac{1}{x}\,dx = \ln | x | + C$ |
| $e^x$ | $\int e^x\,dx = e^x + C$ | ||
| $\sin x$ | $\int \sin x\,dx = -\cos x + C$ | ||
| $\cos x$ | $\int \cos x\,dx = \sin x + C$ |
Integrals also follow linearity. If $a$ and $b$ are constants, then
$$
\int \left(af(x) + bg(x)\right)\,dx = a\int f(x)\,dx + b\int g(x)\,dx
$$
and similarly for definite integrals.
Linearity of integration:
$$
\int \left(af(x) + bg(x)\right)\,dx = a\int f(x)\,dx + b\int g(x)\,dx
$$
Constants can be taken outside the integral.
Geometric Interpretation
The definite integral can be understood visually as the sum of narrow rectangles under a curve. The more rectangles we use, the better the approximation.
Each small rectangle has approximate area $f(x)\Delta x$. Adding all such rectangles suggests
$$
\sum f(x)\Delta x
$$
and the integral is the limiting case of this sum.
Integrals in Physics
Physics often deals with quantities that are distributed continuously. If mass is spread along a rod, charge is spread over a surface, or force changes with position, then the total is found by integration.
Suppose a thin rod lies along the $x$ axis, and its linear mass density is $\lambda(x)$, measured in kilograms per meter. A tiny piece of length $dx$ has mass
$$
dm = \lambda(x)\,dx
$$
Adding all pieces from $x=a$ to $x=b$ gives the total mass:
$$
m = \int_a^b \lambda(x)\,dx
$$
This same pattern appears throughout physics. A small part is written first, then all small parts are added using an integral.
| Physical quantity | Small contribution | Total |
|---|---|---|
| Mass of a rod | $dm = \lambda(x)\,dx$ | $m = \int \lambda(x)\,dx$ |
| Charge on a line | $dq = \lambda_q(x)\,dx$ | $q = \int \lambda_q(x)\,dx$ |
| Area from strip width | $dA = y(x)\,dx$ | $A = \int y(x)\,dx$ |
| Volume from slices | $dV = A(x)\,dx$ | $V = \int A(x)\,dx$ |
At this stage, the key idea is not the specific formula for each topic, but the structure: identify a tiny piece, then integrate.
Units in Integrals
In physics, integrals must make sense dimensionally. The units of the result come from multiplying the units of the integrand by the units of the differential.
For example, if $\lambda(x)$ has units of $\mathrm{kg/m}$ and $dx$ has units of $\mathrm{m}$, then
$$
\lambda(x)\,dx
$$
has units of kilograms, so the integral gives mass.
If velocity $v(t)$ has units of $\mathrm{m/s}$ and $dt$ has units of $\mathrm{s}$, then
$$
v(t)\,dt
$$
has units of meters.
This unit check is a powerful way to see whether an integral setup is reasonable.
In a physical integral, always check units.
Units of result = units of integrand $\times$ units of differential.
Simple Examples
Consider the integral
$$
\int_1^3 2x\,dx
$$
An antiderivative of $2x$ is $x^2$, so
$$
\int_1^3 2x\,dx = \left[x^2\right]_1^3 = 9 - 1 = 8
$$
Now consider a density example. Suppose a rod from $x=0$ to $x=2$ has linear density
$$
\lambda(x) = 3x
$$
in $\mathrm{kg/m}$. Then its mass is
$$
m = \int_0^2 3x\,dx
$$
Evaluating,
$$
m = 3\left[\frac{x^2}{2}\right]_0^2 = 3\cdot 2 = 6\ \mathrm{kg}
$$
The procedure is always the same. Write the tiny contribution, then sum continuously.
Common Ideas to Remember
An integral does not only mean area. Area is one interpretation, but in physics an integral often means total accumulated quantity. The same mathematical operation can represent mass, charge, volume, probability, energy, and many other totals depending on the context.
Also, the variable used in the integral is a dummy variable. For example,
$$
\int_0^1 x^2\,dx = \int_0^1 u^2\,du
$$
These are the same value. The letter inside the integral can change, as long as the meaning stays the same.
Finally, reversing the order of limits changes the sign:
$$
\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx
$$
and integrating over an interval of zero length gives zero:
$$
\int_a^a f(x)\,dx = 0
$$
Important properties of definite integrals:
$$
\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx
$$
and
$$
\int_a^a f(x)\,dx = 0
$$
Closing View
An integral is the mathematics of continuous addition. It allows physics to move from tiny local pieces to whole quantities. Whenever a problem can be described as "add up all the small parts," integration is likely the right tool. The next step in using integrals well is learning how to interpret them physically, and how they connect to rates of change and real physical systems.