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Do you understand the relation between summation and integrals?Abhishek11235 said:
Basic ideas that you start with in Calculus take you from big steps to small steps and then you look at the limit as the step size approaches zero. That sort of relationship can be regarded as as 'continuous'. BUT that doesn't apply to all relationships. Not all relationships or mathematical functions are 'continuous and differentiable' over their whole range and you cannot do simple calculus in those cases. It is lucky (?) that most of the Physics we start off with is amenable to basic calculus methods (differentiation and integration). If you want to get deep into mathematical analysis methods then it will make you able to make choices about when you can and when you can't use basic calculus but, if you are like most of us, you just stick to the 'rules' that they use in book work and you won't go far wrong.Abhishek11235 said:
Summation is the process of adding together a finite number of terms, while integration is the process of finding the area under a curve. In other words, summation deals with discrete quantities while integration deals with continuous quantities.
In many real-world situations, the data or function being analyzed is continuous and cannot be accurately represented by a finite number of discrete points. By changing from summation to integration, we can more accurately model and analyze these continuous quantities.
The process of changing from summation to integration involves taking the limit of a summation as the number of terms approaches infinity. This is known as the Riemann sum, and it can be represented using the integral symbol.
Changing from summation to integration is used in many areas of science and engineering, such as physics, chemistry, and economics. It allows us to accurately model and analyze continuous quantities, such as velocity, acceleration, and population growth.
While integration allows us to more accurately model continuous quantities, it does have its limitations. In some cases, the function being integrated may be too complex or the data may not be continuous, making it difficult to accurately use integration methods. Additionally, the process of changing from summation to integration may be time-consuming and require advanced mathematical knowledge.
