The definite integral for physicists — from Riemann sums to real applications

Physicists integrate many times a day — without thinking about it. Displacement from velocity, work from force, charge from charge density, energy from power, mass from density — they're all integrals. If you only work with constant-acceleration formulas (§2.4), the integral is "invisible" — Newton did it once and hid it inside the formula. But the moment you deal with variable acceleration, position-dependent force, or an arbitrary function, you need the integral directly.

This article explains the definite integral from a physicist's angle — not just pure math. If you haven't taken calculus yet, this is a good starting point.

Why we need this — a motivating example

Scenario: imagine a car whose speed changes:

How far does it travel? For each piece:

Total: 125 m.

That was easy because the velocity was piecewise simple. But what if v(t) = 5 + 3\sin(t)? Not a rectangle, not a trapezoid — a curved shape. That's where the integral comes in.

The core idea — chop into thin rectangles

If v(t) has any shape, we can approximate the area beneath it with many very thin rectangles:

\[ \sum_{i=1}^{n} v(t_i)\, \Delta t \]

This is a "Riemann sum". It's approximate — since the true velocity isn't exactly constant in each piece. But the larger n (equivalently, the smaller Δt), the better the approximation.

The limit — making it "exact"

The definite integral is the limit of the Riemann sum as n → ∞:

\[ \boxed{\int_a^b v(t)\, dt = \lim_{n \to \infty} \sum_{i=1}^{n} v(t_i)\, \Delta t} \]

Geometric interpretation: the exact area under the curve v(t) from t=a to t=b.

Physical interpretation: the displacement of the object during [a, b]. Speed becomes distance.

Notation:

First integral — a constant function

If v(t) = c is constant, the area is a rectangle:

\[ \int_a^b c\, dt = c(b - a) \]

Area = height × width. Simple. This special case is just "distance = speed × time" for constant-velocity motion.

Second integral — a linear function

If v(t) = t (linearly rising with time), the area from 0 to T is a triangle with base T and height T:

\[ \int_0^T t\, dt = \frac{1}{2} T^2 \]

Check by limit. Divide into n pieces, width Δt = T/n. Value at piece i: t_i = i \cdot T/n. Sum:

\[ \sum_{i=1}^{n} \frac{i T}{n} \cdot \frac{T}{n} = \frac{T^2}{n^2} \sum_{i=1}^{n} i = \frac{T^2}{n^2} \cdot \frac{n(n+1)}{2} = \frac{T^2 (n+1)}{2n} \]

As n → ∞, (n+1)/n → 1:

\[ \lim_{n \to \infty} \frac{T^2 (n+1)}{2n} = \frac{T^2}{2}\ ✓ \]

The power rule — the integral calculator

You could work out every function this way, but fortunately there's a general rule for powers:

\[ \boxed{\int t^n\, dt = \frac{t^{n+1}}{n+1} + C \quad (n \neq -1)} \]

where C is the constant of integration (for now, ignore — it drops out in definite integrals). That's the "indefinite integral"; for the definite one:

\[ \int_a^b t^n\, dt = \left[\frac{t^{n+1}}{n+1}\right]_a^b = \frac{b^{n+1} - a^{n+1}}{n+1} \]

Examples:

\[ \int_0^3 t^2\, dt = \frac{3^3 - 0}{3} = 9 \]

\[ \int_1^2 t^4\, dt = \frac{32 - 1}{5} = 6.2 \]

\[ \int_0^T \sqrt{t}\, dt = \int_0^T t^{1/2}\, dt = \frac{T^{3/2}}{3/2} = \frac{2}{3} T^{3/2} \]

A few general rules

1. Linearity: integral of a sum = sum of integrals:

\[ \int_a^b [f(t) + g(t)]\, dt = \int_a^b f(t)\, dt + \int_a^b g(t)\, dt \]

2. Constant factor: a constant pulls outside the integral:

\[ \int_a^b c \cdot f(t)\, dt = c \int_a^b f(t)\, dt \]

3. Direction: swapping limits flips the sign:

\[ \int_a^b f(t)\, dt = -\int_b^a f(t)\, dt \]

4. Splitting the interval: integral over [a, c] = integral over [a, b] + integral over [b, c]:

\[ \int_a^c f(t)\, dt = \int_a^b f(t)\, dt + \int_b^c f(t)\, dt \]

Worked example — a physics application

Scenario: a car starts from rest with acceleration:

\[ a(t) = 2t + 3\quad (\mathrm{m/s^2}) \]

Velocity and position at t = 4 s?

Velocity (from integrating acceleration):

\[ v(4) - v(0) = \int_0^4 (2t + 3)\, dt = [t^2 + 3t]_0^4 = 16 + 12 = 28 \]

With v(0) = 0: v(4) = 28 m/s.

Position (from integrating velocity):

First get v(t) via the indefinite integral: \[ v(t) = t^2 + 3t + C_1 \]

With v(0) = 0, C_1 = 0. So v(t) = t² + 3t.

\[ x(4) - x(0) = \int_0^4 (t^2 + 3t)\, dt = \left[\frac{t^3}{3} + \frac{3t^2}{2}\right]_0^4 = \frac{64}{3} + 24 \approx 45.33\ \mathrm{m} \]

Insight: with the integral tool, even when acceleration varies, you can compute the distance exactly.

Fundamental theorem of calculus (brief)

Differentiation and integration are inverses. If:

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

then:

\[ \frac{dF}{dx} = f(x) \]

That's the "fundamental theorem of calculus" — the most important mathematical result of the 17th century (Newton and Leibniz found it independently).

Practical consequence: if you can differentiate something, you can also integrate it (undo the derivative). Integral tables are essentially "reverse-derivative" tables.

A few integrals worth memorizing

Function Integral
1 t
t^n (n ≠ -1) t^{n+1}/(n+1)
1/t \ln|t|
e^t e^t
e^{kt} e^{kt}/k
sin(t) -cos(t)
cos(t) sin(t)
sin(kt) -cos(kt)/k

Memorize these and you can handle 80% of general-physics integrals.

Practical uses in physics

1. Displacement from velocity (§2.6): \[ \Delta x = \int_a^b v(t)\, dt \]

2. Work from force: \[ W = \int_{x_1}^{x_2} F(x)\, dx \]

3. Gravitational potential energy (Gravity Article 3): \[ U(r) = -\int_\infty^r F(r')\, dr' = -\frac{GMm}{r} \]

4. Mass from density: \[ M = \int \rho\, dV \]

5. Electric charge from charge density: \[ Q = \int \rho\, dV\ \text{or}\ Q = \int \sigma\, dA \]

6. Average of a function: \[ \bar f = \frac{1}{b-a} \int_a^b f(t)\, dt \]

When no closed form exists — numerical integration

Some functions have no closed-form integral. For example:

\[ \int_0^t e^{-t'^2}\, dt' \]

(That's the error function \text{erf} — defined, but with no simple formula.)

In that case we compute numerically:

Rectangle rule: sum of rectangles with small width Δt: \[ \int_a^b f(t)\, dt \approx \sum_i f(t_i) \Delta t \]

Trapezoidal rule: instead of rectangles, trapezoids (connect the left and right point of each piece with a straight line). Better accuracy: \[ \int_a^b f(t)\, dt \approx \sum_i \frac{f(t_i) + f(t_{i+1})}{2} \Delta t \]

Simpson's rule: parabolas instead of straight lines. Much better accuracy with fewer points.

In Python, scipy.integrate.quad() handles all of this automatically.

A few notes and common mistakes

1. Don't forget dt. ∫ f without dt is meaningless. dt tells you "which variable" you're integrating over.

2. Limits must be compatible with the function. If f(t) isn't defined on part of the interval, the integral over that part is undefined. E.g. ∫_{-1}^{1} (1/t²) dt diverges because it blows up at t=0.

3. Track units. ∫ v dt: if v is in m/s and t in s, the answer is in m. Every integral is a unit product — since intrinsically you're summing "value × infinitesimal width".

4. Sign depends on the function. If f(t) < 0 over part of the interval, that part gives negative area. For absolute geometric area, take the absolute value: ∫ |f(t)| dt.

5. Substitution (u-substitution) is not fast to learn but is important. If an integral looks hard, a substitution might simplify it. E.g., ∫ 2t cos(t²) dt with u = t², du = 2t dt becomes ∫ cos u du = sin u = sin(t²). You'll learn this in a calculus course.

What you should be able to do

After this article, you should be able to:

Next steps

If you've mastered this and want to go deeper:

If it's still fuzzy, revisit §2.6 Graphical Analysis, which covers the same idea with more graphical intuition.

📚 A good reference: Calculus by James Stewart (a classic). Free alternative: Paul's Online Math Notes (tutorial.math.lamar.edu). 📖 In physics context: Halliday Vol 1 math appendix, or Feynman Lectures Vol I Ch 8-9.

Have a question? 🤔

If something isn't clear or you have a question, ask it here. The answer will be published on this page.