What is integration please?

  • #1

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Please ,What is integration and how it is a limit and how have they found that integration is the opposite to differentiation and how can we get an area of a curve by integration and what is it's idea also what are the applications of integration specially in physics?
(I studied integration in high school but i am still a beginner i can just solve equations but I don't know where it comes from)
Thanks in advance.....
 

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  • #2
cristo
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  • #3
mathwonk
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given a wire, differentiation is the process of finding its density from its weight everywhere, and integration is the process of finding its weight from its density everywhere.

i got this from mike comenetz's excellent book, calculus, the elements.

for a curve in the plane, differentiation is the process of finding its slope from its height, and integration the opposite.

the integration process also finds the area from the height.

differentiation also finds the area of a cross section of a solid from the volume.

you should understand that such general questions may receive answers that are not too precise.

for functions, integration is also similar to a process of averaging values of the function, and differentiation a process of approximating the change in its value, from a given value, as a multiple of the change in its argument, i.e. of linearizing the function.

e.g. since for small change h in the argument of the function x^2, we have the change in the value as (x+h)^2 - x^2 = 2xh + h^2. since h is small, h^2 is extremely small, the change in the value is approximately 2xh, which is the multiple of h given by the multiplier 2x, so the derivative is the multiplier 2x.
 
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  • #4
mathwonk
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averaging the values of a function means counting them according to how often they are taken on. e.g. if a function equals 3 on the interval [0,2], and equals 5 on the interval [2,5], then we multiply 2 by the length of the interval [0,2], getting 4, and multiply 5 by the length of [2,5] getting 15, and add to find 19 as the integral.

for more general functions f, riemann approximates them by functions like these that are constant on intervals, and takes the limit of those integrals to be the integral of f.
 
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