Q:

How are integration and differentiation connected?

A:

The fundamental theorem of calculus states that differentiation and integration are opposing operations. The definite integration of a function is its antiderivative, obtained through integration and reversed through differentiation. It is called the first fundamental theorem of calculus.

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The second part of the theorem, called the second fundamental theorem of calculus, states that the definite integral of a function is computable using any of an infinite number of antiderivatives. Some information about the original function is lost during differentiation, and reversing the process through integration yields only a general antiderivative expression for the original function.

Although differentiation and integration were well-known before then, their complementary nature was first formally proved by mathematician James Gregory in the 1600s. The fundamental theorem gives a geometric interpretation to differentiation and integration of continuous functions. The area under a curve is found by integrating curve function, while the slope of the curve at any given point is found by differentiating the curve function.

The theorem can also be intuited as the sum of infinitesimal changes of a quantity over time being equal to the net change in quantity. The finer the increments between measurements, the more accurate the information found about how the quantity changes.

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