3.8 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

## 3.8 Applications of calculus (optional calculus-based section)

In section 2.7, 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. 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=t_1` and `t=t_2` is notated like this:

`"area under curve"=Deltax=int_(t_1) ^(t_2)vdt`.

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

`a=(d^2x)/(dt^2)`.

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/s^{2}, and the notation correctly suggests that: the top looks like it has units of meters, the bottom seconds^{2}. The notation is not meant, however, to suggest that `t` is really squared.

3.8 Applications of calculus by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.