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Home > Fundamental theorem of calculus

The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverses of each other. This means that if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence of this, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated.

1 Intuition

Intuitively, the theorem simply says that the sum of infinitesimal changes in a quantity over time (or some other quantity) add up to the net change in quantity.

To get a feeling for the statement, we will start with an example. Suppose a particle travels in a straight line with its position given by x(t) where t is time. The derivative of this function is equal to the infinitesimal change in x per infinitesimal change in time (of course, the derivative itself is dependent on time). Let us define this change in distance per time as the speed v of the particle. In Leibniz's notation:

Rearranging that equation, it is clear that:

By the logic above, a change in x, call it , is the sum of the infinitesimal changes dx. It is also equal to the sum of the infinitesimal products of the derivative and time. This infinite summation is integration; hence, the integration operation allows the recovery of the original function from its derivative. Clearly, this operation works in reverse as we can differentiate the result of our integral to recover the speed function.

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