The accumulation of a rate is given by the change in the amount.

There is a another common form of the Fundamental Theorem of Calculus:

From this you should see that the two versions of the Fundamental Theorem are very closely related. In reality, the two forms are equivalent, just differently stated. Hence people often simply call them both “The Fundamental Theorem of Calculus.” One way of thinking about the Second Fundamental Theorem of Calculus is:

This could be read as:

The accumulation of a rate is given by the change in the amount.

When we compute a definite integral, we first find an antiderivative and then evaluate at the limits of integration. It is convenient to first display the antiderivative and then evaluate. A special notation is often used in the process of evaluating definite integrals using the Fundamental Theorem of Calculus. Instead of explicitly writing , we often write meaning that one should evaluate at and then subtract evaluated at

Let’s see some examples of the fundamental theorem in action.

Understanding motion with the Fundamental Theorem of Calculus

We know that

  • The derivative of a position function is a velocity function.
  • The derivative of a velocity function is an acceleration function.

Now consider definite integrals of velocity and acceleration functions. Specifically, if is a velocity function, what does mean?

The Second Fundamental Theorem of Calculus states that where is any antiderivative of . Since is a velocity function, we can choose to be the position function. Then, measures a change in position, or displacement over the time interval .

Now we know that to solve certain kinds of problems, those that involve accumulation of some form, we “merely” find an antiderivative and substitute two values and subtract. Unfortunately, finding antiderivatives can be quite difficult. While there are a small number of rules that allow us to compute the derivative of any common function, there are no such rules for antiderivatives. There are some techniques that frequently prove useful, but we will never be able to reduce the problem to a completely mechanical process.