Week 10
Modelling accumulation: Integration

Business

The start of the week email. No need to reread it if you received it. It's just linked here in case anyone accidentally deleted it (it's happened before 🥺).

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

Integration 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.

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.

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 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?

We consider one possible crop: canola. It might not be as exciting as growing apples or wine grapes, 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 tempearture by 2 degrees? 4 degrees? 6 degrees?"

1. What is a degree day? 📹 Pencast 8:21 |📘 PDF

Most plants grow according to the heat they accumulate. The units used to measure that accumulation are degree days. Building heating and cooling design is also based on degree days making degree days a surprisingly useful unit of measure. If you haven't encountered them before this pencast will introduce them to you.

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 📹 Pencast 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 📹 Pencast 9:24

Using Excel to actually calculate canola GDD now, and after possible temperature increases due to climate change.

All the exercises in one place

Both due by 4 pm Monday March 15.

Exercise Set 9 Part 1: Integrals as areas Problems and Solutions.

Exercise Set 9 Part 2: Evaluating integrals Problems and Solutions.

Assignment

Assignment 8 is posted. Due by 4 pm Friday March 19.

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 hwo 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 isntead of GDD. Start by sketching the area you want to find.

Logistics

Submit PDF files of work done by hand, or in a word processor. (Let me know if you are unsure how to create these.) Email is preferred, but you can also submit your work through Moodle.
Excel or other binary files should be sent 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 file you submit A8_firstname.pdf, e.g.

A8_Tim.pdf

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

A8_Tim_1.pdf
A8_Tim_2.pdf