On the Applications of Calculus

Among the different fields of mathematics, calculus is most probably the one high school students are most often exposed to, and not for no reason. The field itself deals with continuous variables, which by nature occurs more often than discrete variables as a function of time. Therefore, not only studying this assists a lot for Mathematics-specialized students from doing differential equations and finding limits to solving complex Olympiad-level polynomial problems, but it benefits every many other different areas such as Physics and Chemistry as well. I think a short review, despite an incomplete one, of these applications would probably be a very interesting take.

(Physics) Position/Distance vs. Speed vs. Acceleration as a function of Time:

One of the most basic examples given to calculus beginners, this is by far the most easily comprehensible and most representative of integrals and derivatives. Since speed describes the increase in distance over time, and acceleration describes the increase in speed over time, it makes it clear that speed and acceleration is the derivative of distance and speed with respect to time, respectively. This creates the equation for speed provided constant acceleration a:

and for position:

This means that if we manage to measure just one of the three variables a, v, or x, the other two variables can be calculated as a consequence with a margin of error only dependent on the original measurement, hence reducing the work needed by a huge margin.

Area and volumetric calculations: 

Sometimes my teacher used to say to me that “the ultimate goal of studying calculus is to calculate the volume of a randomly-shaped object”. Of course, that was only a joke, but it reveals quite a famous application dedicated solely to this field of mathematics. For a fixed shape like the cube or the cylinder, obtaining the appropriate formula for calculation might be possible by experimentation, but for a completely random shape it is almost impossible to do so. That is where calculus comes in handy: by “slicing” the object into cross-sections, we changed the calculation of volume into the calculation of area, and by further dividing the area into lines, we changed again the calculation of area into the calculation of length, which is easily measurable. With this interpretation, the area is the integral of length with respect to a length unit x:

and volume is the integral of the area with respect to x as well:

With these formulas, it is possible to approximate the volume of a random object with just length measurements, and the smaller the quantity dxis, the better the accuracy of the volume would be.

 (Chemistry) Rate of reaction:

This is also another very important and heavily studied concept in Chemistry that involves the application of Calculus. The rate of reaction for a particular reactant A is best described as the rate of reduction of the reactant’s concentration, [A], through time, and is portrayed in this formula:

but there is more than just that. Normally, the rate of reaction for the nthorder would be:

Consider the first-order reaction (n = 1). That would leave the following equation:

which is called the integrated rate law of the first-order reaction.

Similarly, we can also find the integrated rate law of reactions of other order – I will not go into detail here. Next time, I will dive a little bit more on logistic differential equations and its applications, but for now – math on!