The Fundamental Theorem of Calculus connects differentiation and integration from "summary" of Skills in Mathematics - Integral Calculus for JEE Main and Advanced by Amit M Agarwal
The Fundamental Theorem of Calculus is a pivotal concept in calculus that establishes a profound connection between differentiation and integration. It essentially states that these two seemingly distinct operations are, in fact, inverses of each other. Differentiation, which involves finding the rate of change of a function, is intimately linked to integration, which involves finding the accumulated total of a function over an interval.
The theorem is divided into two parts, each highlighting a crucial aspect of this connection. The first part states that if a function is continuous on a closed interval, then the accumulation of the function's rate of change over that interval is precisely equal to the difference in the function's values at the endpoints of the interval. This essentially means that integration can be thought of as a way to "undo" differentiation, allowing us to recover the original function from its rate of change.
The second part of the theorem goes a step further, asserting that if we define a new function as the integral of the rate of change function, then this function is continuous and differentiable on the interval. In essence, this part solidifies the idea that differentiation and integration are not just related but are essentially inverse operations of each other.
This profound connection between differentiation and integration has far-reaching implications in mathematics, particularly in calculus. It provides a powerful tool for solving a wide range of problems, allowing us to seamlessly switch between differentiation and integration as needed. By understanding and applying the Fundamental Theorem of Calculus, mathematicians are able to tackle complex problems with greater ease and efficiency, making it a fundamental concept in the study of calculus.
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