Using Integrals in Real Life: A High Schooler's Guide

[RepoMan]

I am a sophomore in high-school and have been teaching myself calculus. I understand it all, but I have a hard time seeing where integrals apply to physics and real life. I get the basic stuff, like using integrals to find area on a graph, and how you can use them when talking about velocity and acceleration. But where else do you use integrals?

 

[Phyisab****]

You can open any book on physics and see integrals on every page.

$$W = \int \vec{F}\bullet d\vec{l}$$

heck, look how many integrals you see on this page

http://en.wikipedia.org/wiki/Maxwell's_equations

 

[stevenb]

I am a sophomore in high-school and have been teaching myself calculus. I understand it all, but I have a hard time seeing where integrals apply to physics and real life. I get the basic stuff, like using integrals to find area on a graph, and how you can use them when talking about velocity and acceleration. But where else do you use integrals?

I agree with the above that integrals are everywhere in physics.

Also, consider that one of the most important principles in physics is Hamilton's Principle, sometimes called "the principle of least action". This principle seems to apply to all physics theories from simple Newtonian Mechanics to the latest string theories. Action is an integral.

http://en.wikipedia.org/wiki/Action_(physics)

http://en.wikipedia.org/wiki/Hamilton's_principle

 

[Phyisab****]

For anyone here to try to give examples of the uses of integration is almost misleading. The uses of integration in physics are so numerous that for us to give you a few examples is not even scratching the surface of the applications of integration.

 

[maverick_starstrider]

When one learns a new area of physics often the biggest hurdle is learning a new area of math. For example:

Classical Mechanics -> Calculus of Variations (Involves integrals of functions of functions)

Electricity and Magnetism -> Vector Calculus (Involves integrals over surfaces and volumes of vector functions)

General Relativity -> Differential Geometry (Involves integrals confined to special types of "surfaces" (called manifolds) in higher dimensional spaces)

Quantum Mechanics -> Partial Differential Equations and Linear Algebra (Partial Differential Equations involves and Enormous amount of integrals with dozens of approximations schemes (Legendre Expansion, Laguerre Expansion, Associated Laguerre Expansion, Hermite Expansion, etc.) all filled with integrals). Also Complex Analysis (The calculus of functions that take complex numbers as inputs (i.e. the have an imaginary component, $$\sqrt{-1}$$))

Quantum Field Theory -> Gaussian Integrals (much of the physical research of QFT has been guided, hampered and pushed forward by progress in solving Gaussian Integrals), Calculus of Variations

This list is certainly not exhaustive and one should definitely not come away from it thinking that in Electricity and Magnetism, for example, one ONLY uses vector calc. All the main calculuses: Real Calculus (the calculus of functions of real numbers), Complex Analysis (the calculus of functions of complex numbers), Functional Analysis/Calculus of Variations (the calculus of functions of functions, so called functionals) are ubiquitous in every area of physics. Those plus a healthy super-sized helping of linear algebra and algebraic theory gets you pretty much all of physics. Suffice it to say integrals are everywhere in physics and knowing how to deal with the different ways they show up is MOST of the heavy lifting for a physicist.

 

[paulfr]

It is not as hard as everyone is making it !

Integration is everywhere in physics because all Integration is
MULTIPLICATION ..... when one of the factors is changing.

In Physics or Math we multiply to find ... Area, Surface Area, Perimeter, Circumference, Volume, Work, etc
If one of the factors is changing we need to Integrate.

Simple examples
1/ 3 x 4 = 12 ... this is the integral of y=3 from x = 0 to x= 4
This is the area under the curve y = 3.
Now just imagine a more complex y like y = x^2. You need to integrate now.

2/ Volume = Area x height now suppose Area is a function of height.
You need to integrate to find V.

Wherever you multiply to find a quantity, you may need to integrate in a more complex situation
with the same variables

 

[chiro]

I am a sophomore in high-school and have been teaching myself calculus. I understand it all, but I have a hard time seeing where integrals apply to physics and real life. I get the basic stuff, like using integrals to find area on a graph, and how you can use them when talking about velocity and acceleration. But where else do you use integrals?

To add what everyone has said, one thing you should realize is that functions can represent any kind of measurable phenomena which are part of any science.

One of the great things about calculus is its Fundamental theorem which states that integration and differentiation are inverse processes and that when an analytic version of the integral of a function over some domain is known then it is simply the difference of one result with another dependent on the domain (ie F(b) - F(a)).

This is extremely powerful.

So basically if we have some system and we know how it changes we can find out what the system will be at any point in time if there is an analytic anti-derivative function.

One other important thing that an above poster has mentioned is that it generalizes the notion of different dimensional measures like length,area,volume etc to objects that have changing measure (ie things that don't have straight lines as their boundaries).

Basically with calculus, anything that can accurately be described as change with respect to something can be analyzed.