Integrals in Physics — Adding Up the Infinitely Small

Adding up the infinitely small

If the derivative asks 'how fast is it changing?', the integral asks 'how much has piled up?' Slice a changing quantity into infinitesimal slivers, add them all — that sum is the integral, written with the elongated S:

$$\int v\,dt$$

Geometrically it is the area under the curve. Physically it is accumulation: velocity accumulated over time becomes displacement; force accumulated over distance becomes work; current accumulated over time becomes charge. One symbol, endless totals.

The two integrals you will use most

The two most-used physics integrals:

Displacement from velocity:

$$\Delta x = \int_{t_1}^{t_2} v(t)\,dt$$

The definite integral between two times is exactly the signed area under the v-t graph — the same area you learned to compute with triangles, now generalized to any curve.

Work from a variable force:

$$W = \int_{x_1}^{x_2} F(x)\,dx$$

When force changes with position (a spring, gravity over large distances), $F \times d$ fails and only the integral gives the true work. For the constant-force case, the integral collapses back to $W = F\Delta x$ — the familiar formula is a special case, not a rival.

Reversing the power rule: +C in action

Integrating a polynomial reverses the power rule: raise the exponent by one, divide by the new exponent, and add $+C$.

$$\int t^n\,dt = \frac{t^{n+1}}{n+1} + C$$

Mini-example: velocity $v = 3t^2$ m/s, starting position $x(0) = 2$ m.

• Integrate: $x(t) = \int 3t^2\,dt = t^3 + C$
• Apply the initial condition: $x(0) = 0 + C = 2$, so $C = 2$
• Result: $x(t) = t^3 + 2$

At $t = 2$ s: $x = 8 + 2 = 10$ m. The $+C$ was not decoration — it placed the motion at the right starting line. Every indefinite integral in physics gets its constant from reality: initial position, initial velocity, or a reference point.

The fundamental theorem: two views, one quantity

The Fundamental Theorem of Calculus ties the two operations together: differentiating an accumulation returns the rate, and accumulating a rate returns the total (up to the starting value). Physics exploits both directions constantly:

• Forward (integrate): acceleration → velocity → position; power → energy.
• Backward (differentiate): position → velocity → acceleration; energy → power.

A spring's story shows the loop: Hooke's law $F = -kx$ integrates to potential energy $U = \tfrac{1}{2}kx^2$; differentiating $U$ with respect to $x$ returns the force (with its sign). Rate and total are two views of one quantity — calculus is the lens that switches between them.

Variables and units

Common mistakes

Real-world applications

• Engineers integrate variable forces (like a stretching spring) to find the work done.

• Cosmologists integrate expansion history to get the age of the universe.

• Electrical engineers integrate current over time to find total charge delivered.

• Probability in quantum mechanics comes from integrating |psi|^2 over regions.