Integral Calculus

In the Applications of Calculus section at the end of the previous chapter, I discussed how the slope-of-the-tangent-line idea related to the calculus concept of a derivative, and the branch of calculus known as differential calculus. The other main branch of calculus, integral calculus, has to do with the area-under-the-curve concept discussed in section 3.5 of this chapter. Again there is a concept, a notation, and a bag of tricks for doing things symbolically rather than graphically. In calculus, the area under the v-t graph between t = t₁ and t = t₂ is notated like this:

The expression on the right is called an integral, and the s-shaped symbol, the integral sign, is read as "integral of . . . "

Integral calculus and differential calculus are closely related. For instance, if you take the derivative of the function x(t), you get the function v(t), and if you integrate the function v(t), you get x(t) back again. In other words, integration and differentiation are inverse operations. This is known as the fundamental theorem of calculus.

On an unrelated topic, there is a special notation for taking the derivative of a function twice. The acceleration, for instance, is the second (i.e., double) derivative of the position, because differentiating x once gives v, and then differentiating v gives a. This is written as

The seemingly inconsistent placement of the twos on the top and bottom confuses all beginning calculus students. The motivation for this funny notation is that acceleration has units of m/s², and the notation correctly suggests that: the top looks like it has units of meters, the bottom seconds². The notation is not meant, however, to suggest that t is really squared.