Math7B – Integral Calculus for Bio

The full details of the logistics of this course, as well as a list of resources and other nuances, can be found in the syllabus, but I’ll provide a brief summary here. The lecture will be held in Skye 170 Monday through Thursday, 8:10–9:30am, starting on July 29. The discussion section will be in Skye 171 on Wednesdays, 11:40am–1:30pm. The final exam will be on Saturday (sorry) August 31 from 8–10am in Skye 170. I’ll host office hours in Skye 284 on Mondays 10am–3pm. If you have questions or comments, email me at .

This course will follow the book Calculus for Biology and Medicine, Third Edition, by Claudia Neuhauser, but you do not need to buy the book for this course. I will give you homework problems each week to think about, but your grade in the course will be determined by assessments you take in discussion which are based on the homework, and by the final exam.

As mentioned in the syllabus, YouTube, and WolframAlpha, and Desmos are great technological resources to help you learn the material for this course. Also, Paul’s Online Notes are generally great, especially the end of Paul’s Calculus I and the beginning of Paul’s Calculus II. Additionally, here is a thread online that is relevant to this course: How is Calculus Helpful for Biology Majors?

Course Outline, Notes, Homework, & Assessments

Here is an outline of the content of the course, along with my personal lecture notes (which I really just write for myself to help me lecture, but might be helpful to students), and links to the homework and the assessment for that week. I’ll post my solutions for the weekly assessments and for the final exam after they are given.

Week Zero – Review of Differential Calculus, Trigonometry, Etc

• Homework
• This homework provides a review of material that will be useful in the course, and also informs students what I think they’ve learned in previous math courses, so they can tell me that I’m wrong.

Week One – Definite and Indefinite Integrals, and the FTC

• Lecture Notes   ·   Homework   ·   Assessment Solutions
• We’ll define integrals as a limit that gives us the area under the curve of the graph of a function, and talk about evaluating definite integrals both numerically and geometrically. Then relate these integrals to (anti)derivatives via the Fundamental Theorem of Calculus (FTC) and we’ll talk about definite versus indefinite integrals, Lastly we’ll use the FTC to calculate the area bounded between two curves that are the each the graph of a continuous function. Here are a few links related to this week’s material.
• The Meaning of the Fundamental Theorem of Calculus
• Why is the area under a curve the integral?
• What does the integral of position with respect to time mean?

Week Two – Interpretations & Applications of Integration

• Lecture Notes   ·   Homework   ·   Assessment Solutions
• We’ll talk about a few more interpretations and applications of integration, like how an integral counts the cumulative change in the value of a function, and how, for example, that can correspond to the bioavailability of a drug in the bloodstream. We’ll talk about how we can find the average value of a function over a length of time. Then we’ll return to geometry a bit to learn how we can calculate the length of a curve and the volume of a solid of revolution using integration. You can see an example of this worked out here:
• The volume of the solids formed by taking the region bounded by $y=\ln(x)$ and $y=2$ in the first quadrant and rotating it about the $x$-axis and the $y$-axis, using both the disk/washer and shell method for each.
• Graphs for which a calculus student can reasonably compute the arclength.

Week Three – Techniques of Integration

• Lecture Notes   ·   Homework   ·   Assessment Solutions
• This week we’ll start practicing the various techniques of integration, including substitution (also called $u$-sub, or change of variables), integration by parts, integrating rational functions by using polynomial long division and partial fraction decomposition. And we’ll talk about the integrals \begin{equation*} \int\tan(x)\,\mathrm{d}x \qquad\qquad \int\ln(x)\,\mathrm{d}x \qquad\qquad \int\sec(x)\,\mathrm{d}x \,. \end{equation*} And here are a few links related to this week’s material.
  • Why do We Need to Learn Integration Techniques?
  • What is Integration by Parts, Really?
  • Example of Evaluating an Integral using Partial Fraction Decomposition
  • UCR Math7B Sample Midterm 1
  • UCR Math7B Sample Midterm 2
  • UCR Math7B Sample Midterm 3

Week Four – Further Techniques of Integration

• Lecture Notes   ·   Homework   ·   Assessment Solutions
• This week we’ll talk about evaluating integrals of functions that are products and quotients of trig functions. Then we’re going to talk about trigonometric substitution, and then evaluating improper integrals. Here are a few links related to this week’s material.
  • List of Interesting Integrals for Early Calculus Students
  • Lesser-Known Integration Tricks

Week Five – Differential Equations and Applications

• Lecture Notes   ·   Homework
• This last week we’ll start looking at ordinary differential equations (ODEs), and see how integration helps us solve separable differential equations. Then we’ll talk about some modelling with differential equations, and equilibrium solutions to differential equations. Basically we’ll start a crash course of UCR’s introduction to differential equations, Math46.
• Page of Formulas for the Final

Final Exam

Notes for When I Teach This Course Again

I Need to Better Explain the Implications of the FTC

I asked this question on the final,

Write down a calculation to verify that \begin{equation*} \int \frac{1}{\sin(x)\cos(x)} \,\mathrm{d}x \;\;=\;\; \ln\left(\tan(x)\right) +C\,. \end{equation*}

and was shocked that only three students responded by taking the derivative of the function $\ln(\tan(x))+C$. Most students tried to evaluate the integral. I didn’t ask them a question exactly like this before on homework or an assessment, but I figured that they had internalized at least the part of the FTC that says taking integrals and taking derivatives are “inverse” operations. But I was evidently wrong. I need to be mindful of this if I teach this class again.

Take an Active Role Advising Students how to Take Notes

Students struggle at learning math. I think this is, in part, because they try to employ the same techniques they use to learn in their other classes to math as well, which doesn’t always work. In particular, I noticed this in their note-taking habits. They’ll meticulously write down everything, even things I explicitly said wasn’t that useful but that they should see worked out once. Even when I put them in groups and have them work on an exercise on the board together, they’ll all take the time to write the exercise carefully in their notes before working. Next class I teach, I need to tell students how to take notes in a math lecture. Like, writing stuff down is important, but thinking about what’s being said, making sense of the lecture, is the most important thing. Here’s some relevant reading:

• Taking Lecture Notes in Lectures on MathOverflow
• Taking Notes vs Doing Math by Henri Picciotto
• In Praise of Lectures by T.W. Körner
• Is there research on the efficacy of taking good notes in math class? on MathEdSE
• Inquiry About My Note-Taking Skill on MathEdSE
• Hand out lecture notes or not? on MathEdSE

And this opens the door to the more general questions: In what ways is learning math different from learning other university topics? To what extent do we need to take the time to coach our students in how to learn mathematics?

Stress the Importance of Definitions and Explanations

I thought I did this quite a bit, but it was probably new to the students to be expected to explain, like in writing, what things are and how things work. I asked them questions on the assessments like,

What is the definition of the definite integral of a function on the interval $(a,b)$? What is the definition of the indefinite integral of a function?

Explain, as if explaining to a friend, why $\int_a^b f(x)\,\mathrm{d}x = \int_a^0 f(x)\,\mathrm{d}x + \int_0^b f(x)\,\mathrm{d}x$.

There aren't too many of these sorts of questions I could have asked in the course; we only had a few definitions, and there were only a few concepts to internalize. Maybe it would've been a good idea to give the students a list of all these sorts of questions at the start of the term and tell them at least one would be on the final. This would put some pressure on them to practice their mathematical writing and their skills at explaining concepts.

Write More Examples when Working Through Techniques of Integration

When I introduce a technique of integration, I should write down a whole bunch of integrals that immediately require that trick, to help students recognize the pattern. On the final, there were way more instances of students trying the wrong technique on very easy integrals than I expected.

Be Sure to Assess for Common Misconceptions

There is a common trend in this course where students don’t fully understand the difference between taking an integral and “simplifying” the integrand. I should be aware of this when designing the final.

Explore Definite Integrals that are Exactly 1

There are few neat areas under a graph that turn out to be exactly $1$. At least the two \begin{equation*} \int_0^1 \ln(x) \;\mathrm{d}x \quad\text{and}\quad \int_0^{\pi/2} \cos(x) \;\mathrm{d}x \,, \end{equation*} and I could use these examples as tools when asking questions that require the student to think of definite integrals as areas.

Learn gnuplot

As an educator, I should learn how to use gnuplot.

Personal Technical Notes

Lastly, just a few technical notes to myself. (1) I should title all of my lectures. The students appreciate this so they can add the titles to their notes for reference to specific lectures. (2) I need to literally send the students a link to the iEvaluation website. I only mentioned that they could go evaluate me at the beginning of a lecture, and only two student responded to evaluations. (3) I should download the CSV file from iGrade.ucr.edu and work with that from the very beginning. That way I can just upload my grades at the end of the term, and reduce the risk of errors/typos. (4) A couple students requested that I put solutions to all the exercises. I should do this for the computational exercises if I teach this course again.