[ScientificBoysClub]

Why do we integrate equations in Math or Physics ?

As far as I know integration/differentiation is really cool. It allowed us to reach greatest pinnacles of discoveries in modern science.it made easier to solve any equation! this is one of my most favorite topic in Math. But, if an equation that explains Physics of Natural phenomena getting integrated or differentiated what is going on actually ?

How does it gives us correct result of all physical things in Nature ?
What is the Magic of this Math ? 

How in the world that this integration/differentiation related to a universal tool to calculate any thing in Physical phenomena ?
Why all laws of Nature agree or happy with it ?

[techmind]

Differentiation reveals the rate-of-change (or instantaneous rate-of-use) of the original quantity or equation.
Integration reveals the cumulative effect of the original quantity or equation.


In the physical world, velocity is the cumulative effect of acceleration (v=u+at), distance covered is the cumulative effect of velocity (s=ut). The first equation covers the case of constant acceleration, while the concept of integration generalises this for any arbitrary time-varying acceleration. The second equation covers the case of constant velocity, while the concept of integration generalises this for any arbitrary time-varying velocity.

Current is the movement of charge, so a charge build-up (eg on a capacitor) is the cumulative effect of a current-flow (Q=It), for a constant current. Integration Q = INTGL (I.t) dt   covers the general case of arbitrary time-varying current.

Power is the rate of use of energy - so power is the derivative of energy-flow, or total energy used is the time-integral of power use.


In earlier stages you'll mostly study integration (cumulative effects) over time, but you can also integrate (sum) over wavelength - eg the total energy output by a lightbulb is the sum (integral) of the energy emitted at each wavelength (colour).
You also commonly intgrate (sum) over space or direction, eg the total light output by a bulb is the sum of the light emitted in each direction (it will not in general be uniform in all directions) - although you're rather less likely to have a nice equation for this one!!!


It's just a natural, reasonable (even obvious) thing when you "get" it.  []

[Soul Surfer]

Techmind has said most of it very effectively.  However just let me get a little bit more basic in what he is talking about.  Simple arithmetic and algebra deals with things that have fixed values and are steady, but real life is not like that.  Most things change and differential and integral calculus allows us to analyse things that are changing with time or position or any other factor.

[lyner]

There is something a bit magic in the fact that natural processes can be represented symbolically by Maths and that manipulating the symbols in an abstract way can give accurate predictions of reality.

Of course, there are many examples of physical situations which can be expressed in mathematical terms but the resulting equations ( very often differential equations) cannot be solved analytically but require brand new functions to be defined and the results can only be found by numerical methods. For instance, it is easy to write an equation which describes the forces on a circular membrane (like a drumskin) but you need to invent the Bessel Function which has no 'closed form' solution.
In fact if you think of any random mathematical function, it is very likely that you won't be able to integrate it analytically. The Maths doesn't take you all the way there.

[lightarrow]

Just to add a 'pictorial' consideration to the OP's question.

Do you know how the MPEG (for example) algorithm works in encoding a movie's digital informations? Essentially , it subdivides the movie into many pieces, then it registers the first image and only the "variations" of that one, finding the simplest mathematical law which relates these variations. Then, just with the initial image and that mathematical law, the algorithm is able to reproduce all the movie.

It is the same to what we do with an unknown function in physics: we have a mathematical law and the initial value(s) of that function, and we are able to reproduce all of the function, in practice evaluating all the 'variations' of it.

The 'subdividing' corresponds to the derivation; the inverse, that is the adding of all the pieces together, corresponds to the integration.

So, it's nothing more than that: an ingenious creation of the human mind to simplify the reality.

[ScientificBoysClub]

Differentiation reveals the rate-of-change (or instantaneous rate-of-use) of the original quantity or equation.
Integration reveals the cumulative effect of the original quantity or equation.


In the physical world, velocity is the cumulative effect of acceleration (v=u+at), distance covered is the cumulative effect of velocity (s=ut). The first equation covers the case of constant acceleration, while the concept of integration generalises this for any arbitrary time-varying acceleration. The second equation covers the case of constant velocity, while the concept of integration generalises this for any arbitrary time-varying velocity.

Current is the movement of charge, so a charge build-up (eg on a capacitor) is the cumulative effect of a current-flow (Q=It), for a constant current. Integration Q = INTGL (I.t) dt   covers the general case of arbitrary time-varying current.

Power is the rate of use of energy - so power is the derivative of energy-flow, or total energy used is the time-integral of power use.


In earlier stages you'll mostly study integration (cumulative effects) over time, but you can also integrate (sum) over wavelength - eg the total energy output by a lightbulb is the sum (integral) of the energy emitted at each wavelength (colour).
You also commonly intgrate (sum) over space or direction, eg the total light output by a bulb is the sum of the light emitted in each direction (it will not in general be uniform in all directions) - although you're rather less likely to have a nice equation for this one!!!


It's just a natural, reasonable (even obvious) thing when you "get" it.  []

Thanks a lot what do you mean by the term  cumulative effects ? sorry I am new to it !