How integration can find the summation of infinitely many rectangles?

In summary, definite integration involves finding the area under a curve by dividing it into an infinite number of rectangles and summing their areas. However, this does not mean that an infinite number of things are actually being added up. Instead, the integral is defined as a limit of sums of finite numbers of rectangles, and other types of integrals exist as well. It is important to have a good understanding of limits in order to understand the definition of integration.
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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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Habib_hasan said:
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.
 
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andrewkirk said:
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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1. How does integration find the summation of infinitely many rectangles?

Integration is a mathematical process that involves finding the area under a curve. By dividing the area into smaller and smaller rectangles, and then summing up their areas, integration is able to approximate the total area under the curve, even if there are infinitely many rectangles.

2. What is the significance of finding the summation of infinitely many rectangles?

Finding the summation of infinitely many rectangles is important because it allows us to calculate the area under curves and solve a wide range of problems in fields such as physics, engineering, economics, and more. It is also a fundamental concept in calculus and essential for understanding more complex mathematical concepts.

3. Can integration find the exact value of the summation of infinitely many rectangles?

No, integration can only approximate the value of the summation of infinitely many rectangles. The more rectangles that are used in the approximation, the closer the result will be to the exact value, but it will never be exactly equal. This is due to the nature of infinity, which is impossible to reach or calculate in its entirety.

4. Are there different methods of integration for finding the summation of infinitely many rectangles?

Yes, there are several methods of integration, including the fundamental theorem of calculus, Riemann sums, and the trapezoidal rule. Each method has its own strengths and weaknesses and is used in different situations depending on the complexity of the problem and the desired level of accuracy.

5. Can integration be used for other shapes besides rectangles?

Yes, integration can be used to find the area under curves of various shapes, not just rectangles. This includes circles, ellipses, and even irregular shapes. The process is the same, but the method of dividing the area into smaller parts may be different depending on the shape.

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