Our learning goals for the Applications of Definite Integrals unit in calculus are the following:

I can calculate and use the area between two curves.

I can use the disc and washer methods to calculate and use the volume of a solid.

I can use the shell method to calculate and use the volume of a solid.

I can calculate and use volumes of solids created by known cross sections.

During the lesson focusing on the first goal, we used a scenario from a TImath activity The Area Between to start our conversation.

I rarely send the TNS documents as is to my students or give them a copy of the printed student handout (even though I learn from both in my own planning of how the lesson will play out). This activity gave the following information on the first two pages:

Suppose you are building a concrete pathway. It is to be 1/3 foot deep.

To determine the amount of concrete needed, you will need to:

- calculate area (the integral of the top function minus the bottom function

- calculate volume (area multiplied by depth)

The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by

f(x)=sin(0.5x)+3

g(x)=sin(0.5x)

On the next page, graph the functions. Use the Integral tool to calculate the area under f1 and f2. Then, use the Text and Calculate tools to find the volume of the pathway.

Which takes away any opportunity for students to engage in **productive struggle**.

I shared this instead:

Suppose you are building a concrete pathway that is to be 1/3 foot deep. The borders for the pathway can be modeled on the interval -2π ≤ x ≤ 2π by f(x)=sin(0.5x)+3 and g(x)=sin(0.5x).

(I’m fully aware that giving them even this much information takes away from the modeling process … but there is always give and take, and for this lesson, the learning goal wasn’t whether they could determine functions for modeling the sidewalk.)

They decided to graph the functions.

And talked about how they could calculate the area between the curves.

They had never used the Integral tool for graphs, much less the Bounded Area tool, so they oohed and aahed gasped in amazement.

Sydney asked: Is that the only way to get the area between the curves?

(I knew that she was **looking for and making use of structure**, composing and decomposing the sidewalk into regions with equal area).

I answered: Is it?

We made Sydney the Live Presenter, and she used the Integral tool to calculate the area between f(x)=sin(0.5x)+3 and the x-axis from -2π to 2π.

So how can we calculate the amount of concrete needed? The integral and bounded area tools are helpful for visualizing what you’re calculating, but you can’t use those tools on the AP Exam.

And so the students decided to calculate the area between the curves and then multiply by 1/3 to get the volume of the pathway.

Because they were able to tell me what to do, I almost didn’t send a Quick Poll to collect a definite integral that would calculate the volume. I wanted to hurry up and get to a card-matching activity similar to Michael Fenton’s that I knew would be helpful, but instead I eased the hurry syndrome and sent the poll.

What I saw and heard was well worth the time that it took.

Can you spot the students’ misconception?

Several students were multiplying the definite integral by 4π and by 1/3, to represent height times base times depth, instead of recognize that the definite integral represented height times base (area), and not just height. (They knew this … we had summed the areas of an infinite number of rectangles for a certain base to calculate area under the curve. But they obviously didn’t know this like they needed to.)

When we calculated their integral, we didn’t get (1/3)*37.699, as expected.

Next I purposefully choose a region for which the upper and lower boundaries changed.

We had a nice **look for and make use of structure** discussion about different ways to write a definite integral for calculating the area of the region.

Many of you might notice that there is more opportunity to **look for and make use of structure** for the concrete pathway. I never asked whether you really need calculus to calculate the volume of the pathway. Nevertheless, I feel like I found two good problems/items/tasks to push and probe student thinking. And there’s always next year, as the journey continues …