Week 8
Modelling accumulation: Integration

Business

The start of the week email.

⦿ Monday: Integration, areas and antidifferentiation

Integration as area: Salmon and solar power Pencast 14:53 |PDF
Integration measures the net area between a function and the x-axis. If the function measures a physical quantity, then this area is the total of the quantity. This pencast suggests why this is true.
Integration is antidifferentiation Pencast 8:45 | PDF
The First fundamental theorem of calculus tells us that the value of an integral over an interval is the difference between the values of its antiderivative at the endpoints.
Examples Pencast 11:16 |PDF
Three simple examples of integration.
Examples with sin x Pencast 9:07 |PDF
Two examples of integrating sinusoids.
Integration and differentiation are inverse operations Pencast 7:46 |PDF
Integrating and differentiating are inverse processes, like squaring and square rooting, or exponentiating and taking logs.
⇒ If you need to recover a function from its derivative you integrate.
⇒ If you need to recover a function from its integral you differentiate.
The indefinite integral Pencast 5:05 |PDF
A function's integral is not unique, i.e. each function has an infinite number of integrals (but they differ only by an arbitrary constant). The general formula for the antiderivative is called the indefinite integral.
Sums vs accumulations Pencast 10:11 |PDF
Sums apply to discrete functions. Integrals to continuous functions. There's some notation for sums of discrete functions you should be familiar with (if you aren't already), because you'll encounter it eventually.

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources (both take a more traditional formal approach to the topic of integration)

In Calculus Volume 1 by Gilbert Strang and Edwin Herman
§5.2 covers basic integration.

In Calculus: Early Transcendentals by D. Guichard
Chapter 6 covers basic integration.

⦿ Wednesday: Integration theorems

Integration theorems Pencast 9:11 |PDF
So far we've been finding our integrals using our intuition. Here are a few theorems to put us on a more solid footing. And a warning about two intuitive things that are tempting to do but incorrect!
Handout: Calculus integration formulas PDF
A one-page summary. Print it out, or have it on your screen when you work on the exercise or assignment problems.
More examples Pencast 20:19 |PDF
Four intermediate examples.
Extended example: Penicillin absorption Pencast 25:38 |PDF
Finally we use integration for something real: modelling the absorption of penicillin into the body.

Exercises

If you're comfortable with the topics above you're ready to work on:

Supplementary resources

Same as for Monday:

In Calculus Volume 1 by Gilbert Strang and Edwin Herman
§5.2 covers basic integration.

In Calculus: Early Transcendentals by D. Guichard
Chapter 6 covers basic integration.

⦿ Friday. A case study: Can canola be grown in Whitehorse now? In the future?

Today's question: Can we grow Canola, which requires 1,041 degree days to mature, in Whitehorse? What if climate change warms us up by 2°C? 4°C? 6°C?

Scientists predict that as the climate warms arable areas will move northward, and southern areas will desertify. What might this mean for Whitehorse? Could we become a farming area?

Let's consider one possible crop: canola. It might not be as exciting as growing apples or grapes for wine, but it is a practical cash crop producing healthy cooking oil (the common cooking oil with the least saturated fat), and high-protein meal (80% of the protein of soy meal).

Our plan:

  1. Figure out what the heck a degree day is.
  2. How many degree days does canola need?
  3. Gather annual weather data for Whitehorse.
  4. Use Desmos to fit a sinusoidal model to it.
  5. Find the integral formula for that model.
  6. Use Excel to evaluate that definite integral (because it gives us easy to edit formulas, and a clear record of our calculations.)
  7. Answer the question about canola!
  8. Use the model to answer other what-if questions, e.g. "What if climate change raises Whitehorse's average temperature by 2 degrees? 4 degrees? 6 degrees?"
1. What is a degree day? Pencast 8:21 |PDF
Most plants grow according to the heat of their environment. The units used to measure that accumulation are degree days. Building heating and cooling design is also based on degree days making them a surprisingly useful unit of measure. If you haven't encountered them before this pencast will introduce you to them.
2. Canola and growing degree days (GDD) Pencast 3:53 |PDF
The details about the development of canola by degree days.
3. and 4. Curve fitting our model for Canola GDDs Screencast 7:48
Fitting a curve to Whitehorse's temperature profile.
5. Definite integral for Canola GDDs Pencast 13:59 |PDF
Integrating between the curve and the reference temperature to find a formula for canola GDD.
6–8. Using Excel to evaluate the Canola GDD integral and answer our questions Screencast 9:24
Using Excel to actually calculate canola GDD now, and after possible temperature increases due to climate change.

Exercises

There are no separate exercises for this case study. See Problems 5 and 6 on the assignment for extensions of it.

Supplementary resources

None for this topic.

Assignment

Assignment 8 is posted. Due by 1 pm Wednesday March 18.

About the problems:

1-3. Mechanical problems: Just find the definite integral, substitute the end point values, and calculate.

4. The only "trick" here is figuring out how to find the area between functions, but that's done in our case study this week.

5. Repeat Friday's case study, but using new temperature data.

6. A variation on Friday's case study because we want HDD instead of GDD. Hint: Start by sketching the area you want to find.

Logistics

Excel or other binary files should be emailed as file attachments.
Google sheets can be submitted by sharing a link to them.
Work done in Desmos can be submitted by sharing a link to your saved graph.

Name the files you submit A8_firstname.ext, e.g.

A8_Tim.xlsx

If you have to submit multiple files, add a suffix number to each, e.g.

A8_Tim_1.pdf
A8_Tim_2.pdf