How integration can find the summation of infinitely many rectangles?

  • #1

Main Question or Discussion Point

In definite Integration simply we find out the area under the curve.If I am not wrong the processing is like that we divide the area in infinite numbers of rectangles and then find out the summation of all rectangles area.
My Question is here, how it is possible to add infinity? assume you have a box and there is "n" numbers of marbles. If I ask you to tell me the summation of all of them, is it possible to answer? definitely not. Then without knowing the numbers of rectangles how the integration system can tell the summation of areas of infinity rectangles?
 

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  • #2
andrewkirk
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If I am not wrong the processing is like that we divide the area in infinite numbers of rectangles and then find out the summation of all rectangles area.
That is a common misconception. At no point in an integration is an infinite number of things ever added up.

An integral is defined as a limit of a set of sums of areas, called Riemann sums, where each sum is obtained by dividing the area into a finite number of rectangles. To understand the definition you need a good understanding of limits, which are fundamental to calculus. The definition of this type of integral, called a Riemann Integral, is given here.

There are other sorts of integrals, such as Lebesgue and Stieltjes integrals. But it's easiest to start with Riemann integrals. In any case, none of the definitions require adding up an infinite number of things. They all use limits of sums of finite numbers of things, in one way or another.
 
  • #3
That is a common misconception. At no point in an integration is an infinite number of things ever added up.

An integral is defined as a limit of a set of sums of areas, called Riemann sums, where each sum is obtained by dividing the area into a finite number of rectangles. To understand the definition you need a good understanding of limits, which are fundamental to calculus. The definition of this type of integral, called a Riemann Integral, is given here.

There are other sorts of integrals, such as Lebesgue and Stieltjes integrals. But it's easiest to start with Riemann integrals. In any case, none of the definitions require adding up an infinite number of things. They all use limits of sums of finite numbers of things, in one way or another.
Thanks a lot. I will think and review my textbook again. Actually I am new learner of calculus. If I face any problem again hope you will help me again. [emoji3] [emoji4]
 
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