# Activities for biology undergraduates taking integral calculus

After searching for applications of calculus for biology students, I've found that many of the results are all either contrived exercises, or are way over the heads of students that are seeing calculus for the first time (like activities for a computational biology or biological modelling course). The results that are at an appropriate level all seem to center around differential calculus, and especially related rates.

What are some good activities to give to biology students in a one hour discussion section in an integral calculus course? I would appreciate either specific activities or problems, or just good resources for activities. Here are a few good sources I've found so far:

• Integration Applications (dead link) by R. Vandiver, although these exercises read like, "here's some biology background, now perform this mildly related calculation just because," which I don't think is very engaging.

• Mike Mesterton-Gibbons' Lectures on calculus for life science majors, which I haven't read through too thoroughly yet.

• Similar question: "What are some activities/projects I can assign to calculus students from bio/chem/physics majors to specifically motivate their interest?" matheducators.stackexchange.com/q/1164/80 – Brendan W. Sullivan Jul 24 '17 at 15:58
• Related question: "How is calculus helpful for biology majors?" matheducators.stackexchange.com/questions/2060/… – Brendan W. Sullivan Jul 24 '17 at 15:59
• I'm not convinced that it's interesting to discuss integral calculus applications as something separate from differential calculus applications. In most applications, we have two variables, A and B, where A is the derivative of B and B the integral of A. You can give either A or B, which makes it either an integration problem or a differentiation problem. – Ben Crowell Jul 24 '17 at 21:21
• @BenCrowell If the application involves numerical integration, then I think it is interesting. Most first chapters on integration usually have the Riemann sum definition or the right-hand or left-hand rules for approximating areas under curves. – ncr Jul 30 '17 at 1:56
• Integration is fundamental to probability and statistics, which in turns is fundamental to scientific thought and methods. Integration also allows you to solve differential equations, calculate averages, etc. There's a whole field called mathematical biology and you can't access it without some basic math. Some interesting applications might be too advanced, but you can still find a lot of motivation in this field. – Olivier Aug 3 '17 at 11:46

## 5 Answers

AUC (Area Under the Curve) plays an important role in pharmacokinetics. The basic ideas should be both accessible and interesting to biology students. There is a potentially interesting tidbit associated with this. The trapezoidal rule is often used to estimate this integral, but the much-cited paper which helped to popularize the trapezoidal rule in pharmacokinetics was written by a researcher who didn't realize that they were reinventing a wheel which had been known for centuries.

• This just shows that all these math-hating students who think they will never need math beyond basic algebra in their lives should be forcefully taught at least some basic calculus without regard for fun or immediate usefulness. Also, this shows that grade school does not teach students to work with sources, although in the age of online search engines this should matter less (the "reinventing the wheel" paper appeared in pre-Internet time). – Rusty Core Aug 5 '19 at 18:03

For many years Dorothy Wallace has taught a course at Dartmouth College

https://math.dartmouth.edu/~m4s17/

that gives bio majors and pre-med folks an opportunity to contribute to the mathematical modeling of biological systems literature (predator-prey models, population models, disease models, pharmacokinetics). For many of the projects and assignments in the course, all the calculus a student needs is an intuitive understanding of the derivative, an appreciation for exponential growth and decay and a willingness to use and understand Euler's method. The latter is not typically a Calculus I topic (maybe it should be), but I find that it is not hard for students to pick up. The projects in the course often take the form of "here's an interesting paper or model in the literature, let's consider this tweak to it..."

The course is for students who have completed a semester of calculus, but the population modeling and pharmacokinetics models I feel should be accessible to students taking a first course in calculus. For example: the drug nifurtimox is used to treat chagas, a tropical parasitic disease. You could give them the simple system of differential equations that represents the way the drug is processed by the body (and ask them to explain it) and then ask them which is better: a three dose a day regimen or a four dose a day regimen (the Centers for Disease Control says both regimens are possible but doesn't say which is "better").

Full disclosure: Dorothy and I have a book on this topic: "Applications of Calculus to Biology and Medicine: Case Studies from Lake Victoria" published by World Scientific.

You could study the geometry of a nautilus, whose shape is close to a logarithmic spiral. The nautilus uses its volume ratio of air/water buoyancy to descend and rise. Calculating a nautilus's volume could be an interesting integral calculus exercise. Of course you could start in 2D with the enclosed area. (Image from Wikipedia article.)
And there is quite a bit of (more advanced) calculus in understanding buoyancy, deriving Archimedes' principle.

I think a common example is cardiac output. Here, with the dye dilution technique, you have a monitor that measures dye pushed through the bloodstream. Maybe, for example, you would like to investigate Turkey's heart rates as they run on treadmills. The monitor would measure a concentration in the blood at certain intervals. We can consider the cardiac output as the total volume of dye measured divided by the time as follows:

$$R=V/T$$

Similarly, we can express this as the amount of dye(D) over the volume(CT) as

$$R=D/CT$$

Hence, in our example above, the CT is the sum of the concentrations of dye in the blood, which would be found by integration. I use these examples early in integration before much work with a definite integral.

A specific example could have 5 liters of dye injected with the following concentrations measured every second:

$$c = [0, 0.1, 0.2, 0.6, 1.2, 2.0, 3.0, 4.2, 5.5, 6.3, 7.0, 7.5, 7.8, 7.9, 7.9, 7.9, 7.8, 6.9, 6.1, 5.4, 4.7, 4.1, 3.5, 2.8, 2.1, 2.1, 2.2]$$

Our cardiac output would be found by:

$$R = \frac{5}{\sum_{i=1}^{25} c_i} = \frac{5}{45.43}$$

Graphically, you can interpret this as area approximations under the curve. My advice would be to think broadly about what is relevant for biologists and include some things that are environmental, chemical or even geological.

There are also some biological problems that are purely biological (anything involving depletion or accumulation, biofermenters, toxins, etc. are a good example).

In particular, emphasize examples from human physiology as most biologists are pre-med, pre-pharma, etc. And it is just intrinsically interesting.

Here are some quick Google research results:

http://www.brynmawr.edu/math/people/vandiver/documents/Integration.pdf

https://www.google.com/search?q=integral+calculus+biology+problem&sourceid=ie7&rls=com.microsoft:en-US:IE-Address&ie=&oe=

(Note, I was going to post the actual results but...there is a social justice problem stopping guests from posting more than 2 links.)

• maybe posting as a guest is not a wise thing to do regularly. – James S. Cook Jul 25 '17 at 5:03
• Agreed; you may as well make an account since every regular user of the site recognizes you anyway :) – Chris Cunningham Jul 27 '17 at 14:16