Why do Integrals Work With Discrete Quantities?

Why do we apply Calculus in Physics when most of the quantities are not continuous and are not symmetrical at all levels of magnification ? Aren't most, if not all, forms of Matter and Energy discrete ? We talk about differential elements of energy, charge, fields, current, liquids and so many other quantities which don't or can't exist and we derive results from such assumptions. How is this right ?
- Pawan (age 18)
Chennai, Tamilnadu, India.

Hi Pawan,

I should first point out that many objects in quantum theory are, in fact, continuous variables (i.e. an electron's spatial wavefunction, or a free particle's energy spectrum). In this case, operations (e.g. the dot product) are defined exactly using integrals.

If you are worried about things like the Planck length or time, then you'll have to ask someone a lot smarter than myself. However, I can still tell you why integrals work just fine.

Hopefully your calculus class covered Riemann sums. The integral can be defined as a Riemann sum with infinitely many infinitesimal rectangles. In physics, as you say, some quantities can't be divided infinitely. However, if you are considering effects on a scale much larger than one quantum, then the Riemann sum still has a lot of rectangles, and is a great approximation.

If you care about effects on the scale of the discretization, then you might not be able to use an integral in the usual way. Luckily, when we use approximations, we can often estimate how accurate they are, and deal with the error accordingly.

Hope that makes sense,

David Schmid

(published on 06/03/2013)