What is the Difference Between Integration and Differentiation?

Integration and differentiation are two major concepts in calculus that serve different purposes. Here are the main differences between them:

1. Purpose: Differentiation is used to find the rate of change of a function with respect to a variable, while integration is used to find the area under the curve of a function or to add small and discrete data that cannot be added singularly.
2. Relationship: Differentiation and integration are inverses of each other, meaning that the process of finding the derivative of a function (differentiation) can be reversed to find the original function (integration).
3. Derivatives and Integrals: Derivatives are considered at a point, and they represent the instantaneous rate of change of a function. In contrast, definite integrals of functions are considered over an interval, and they represent the area under the curve of a function over that interval.
4. Uniqueness: The derivative of a function is unique, meaning that there is only one possible derivative for a given function. However, the integral of a function may not be unique, as the value of the integration constant C is arbitrary and can vary between two integrals of the same function.
5. Geometrical Interpretation: The derivative of a function describes the rate of change of the function with respect to a variable, which can be visualized as the slope of the function at a point. On the other hand, the integral of a function represents the area under the curve of the function, which is a more complex geometric concept.

In summary, differentiation and integration serve different purposes in calculus, with differentiation focusing on finding the rate of change of a function and integration focusing on finding the area under the curve of a function or adding small and discrete data. These concepts are inverses of each other, and they have unique properties in terms of their derivatives, integrals, and geometrical interpretations.

Comparative Table: Integration vs Differentiation

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