# The intuitive meaning of integrals

I am an engineering student and i always encounter problems that needs integrals I know that integral is area under the curve , etc.... but till now i could not develop and intuitive meaning for integration. does integration rely only on the idea of area under the curve. do the physics laws that are based on integration like for example the that the work is the integral of the F in infinitesimal distance was proved by the idea of the area under the curve or is there an intuitive way that i could understand the integration through it?

• good choice is to think of averaging process. – tp1 Nov 10 '18 at 3:13
• Just curious, why isn't the idea of area very intuitive? It is something visual, and visual is often intuitive? – Ovi Nov 10 '18 at 3:18
• Well, area under the curve is calculated by really, really thin rectangles (or trapezoids) and the height of those depend on the rate at which the curve change values. – fleablood Nov 10 '18 at 3:20
• Isn't the concept of work intuitive enough? It is even used to give intuitive meaning to contour integrals in complex analysis... – Yuriy S Nov 10 '18 at 8:23
• You may want to look at the often-cited-here book Calculus Made Easy by Silvanus P. Thompson. – Dave L. Renfro Nov 10 '18 at 9:32

## 4 Answers

The integral is basically the total change found from the rate of change.

The way I like to see it is when I look at a package of printing papers, say 500 papers on top of each other, it reminds me of the volume of the package found by slice method namely the integral of the area is the volume.

It is related to the area under the curve where the curve is positive.

For example if you have the rate of money flow you can integrate to find the total money flow.

If you have a formula for the speed you can integrate to find the total distance traveled, or the arc-length.

For intuition from a physics point of view, I would probably not suggest thinking of integrals as areas unless the problem was actually about area. An obvious intuitive interpretation of integration is reversing differentiation. Suppose you know the velocity of a car over a period of time, how do you figure out where it is now? Integrate your data. What about that annoying constant that you get when you integrate? Well, that corresponds to needing to know where the car was at time 0.

For a real example, look at pre-GPS inertial guidance systems. They measure acceleration and perform double integration to calculate position. Inertial navigation system

Measure the rate of energy going into a system, e.g. the voltage and current of an electrical supply, how do you know the total energy? Integrate the power.

Integrals represent the area under a curve as you already know, but the area means a lot of things. It mainly represent an area (obviously) but also the sum of a change (defined by a derivative). That is why finding an integral of a function f(x) is equal to finding a derivative that will give you f(x).

There are lots of formulae from beginning physics and other areas that multiply two things together to get a third thing. Like Area=length x width. Force = mass x acceleration. Distance = rate x time. These formulae are easy to apply if the two factors are constant. Note that you can represent any of them as the area of a rectangle.

But what do you do if one of the factors starts to vary? You have a rectangle, but with one side all wiggly. If the rectangle represents mass x acceleration and the acceleration keeps changing, then the area under the wiggly curve is still the force.

Integration cuts the wiggly "rectangle" into little tiny real rectangles, computes mass x acceleration for each one, and then adds them up to get the whole force. Integration means "whole-ification."