# Wait, the Interval Width Varies?

## Objective

SWBAT compute and interpret definite integrals with variable upper limits algebraically and graphically

#### Big Idea

How can we interpret integrals with function limits of integration? Tables & graphs supplement symbolic representations for understanding.

## Warm-Up + Homework Review

25 minutes

Last night’s homework solutions appear in the In The Classroom file. As class begins, my students will begin working on today's Warm-up problems:

1)  Find the area of the trapezoid under  from n = 3 to n = 7.  Then find the height of the rectangle with the same area as the trapezoid on this same interval.

2)  Study flashcards – warm-up quiz…

Warmup #1 revisits average value. When we go over this task, I will briefly show the Morph Average Value.ggb applet to remind students of the geometry of rearranging the area under the function into a rectangle with same base length.

I don't expect that Warmup #2 will take my students very long. I plan to display a few flashcards, calling on students to verbally respond to them.  Over the course of the year, I find that regular, brief forms of accountability like this go a long way to ensure students are routinely studying their flashcards/programs to retain command of the skills and concepts that support greater depth of understanding of calculus.

## Setting the Stage

15 minutes

I begin today’s instruction with a review of different ways we think about integrals in this class.

• We can interpret a definite integral as the “change in” an accumulated quantity over an interval.  For example, integrating velocity gives the “change in position” or “displacement,” whereas integrating the absolute value of velocity or “speed” gives “total distance” traveled.
• Harking back to yesterday's lesson, integrals with x as a limit of integration result in another function, not a constant value representing the area under a function since the interval’s width can vary with x.

The Investigation section of today's lesson delves further into modeling with integrals (MP 4). We will interpret definite integrals with less-trivial functions as upper limits of integration. So, it is important for students to be comfortable thinking about integrals with variable limits of integration as area accumulators over intervals that vary in width.

Building on our opening conversation I display the original Trace Area under the Curve GeoGebra applet. I ask students to recall how the upper limit of integration x acts as an “area accumulator” as the interval varies in width.  I will trace out the integral functions for at least one of the pre-programmed functions in the applet; we will come back to this traced function in a few minutes.  Then, I will ask students, “how would this applet change if the upper limit of integration became 2x?”

It is important to give students ample time to contemplate this question individually, and then with neighbors. I want students to have sufficient time to analyzing how the accumulated area will change as a result of changing the variable limit of integration in a certain way. It is an excellent opportunity for students to engage in Mathematical Practice 2: reason abstractly and quantitatively. I have my students make predictions for how the applet will behave. This activity has two major benefits:

1. Predictions put students in the position to form conclusions based on observations and their prior knowledge.
2. Predictions create intrigue and interest for students to discover whether their predictions were correct, and then to reflect on ways to make more accurate predictions in the future.

Extending Prior Knowledge
Once students have considered how they think the applet will change and arrived at a prediction, I open Trace Area under curve v2.ggb. This GeoGebra applet consists includes the same functions as the original applet, except the upper limit of integration has changed from x to 2x

Instead of telling students what happens, I will drag the slider of x-values and let students notice what changes.  In this case, changing x to 2x has the effect of widening the interval twice as fast as before; students should notice that the x-coordinate of the green point is twice as large as the x-value on the slider, which shows how the interval is growing in width twice as fast as before. I find that it is helpful to leave the lower limit of integration at 0 to make this phenomenon more readily apparent.

Differentiating the Lesson
I may choose to move the red point to investigate its effects graphically if students are doing well with this topic. When doing this, I use the applet to trace the new integral function for the same function as I traced in Trace Area under curve v1.ggb a few minutes ago. I usually toggle back and forth between these two windows and ask students to notice what has changed about the integral functions.

Typically, students will say something like “the new function is narrower”. I press students to be precise (SMP #6) in describing this narrowness mathematically.  When integrating the function and substituting the upper and lower limits of integration, substituting the 2x for the variable has the effect of a horizontal compression by a factor that is consistent with students’ prior experiences with transforming functions.

## Investigation

15 minutes

Building from students’ prior work with integrals with x as a limit of integration, we now consider integrals with other functions as limits of integration.  The Variable limits FTC exploration worksheet is designed to be a self-directed investigation. As they work, students use graphs and tables to generate the symbolic form of the integrated function and then reason (MP2) about ways to compute the integrals algebraically without using tables and graphs.

Teacher's Note: See Heather Johnson's Mathematics Teacher article from Feb 2010 titled Investigating the Fundamental Theorem of Calculus for detailed discussion about the this activity.

## Closure + Homework

15 minutes

As we prepare for the end of today's lesson, I ask students to write 1-2 sentences and draw a picture detailing why the lower limit of integration represents a vertical shift in the integral function.  The written aspect of this closure prompt is important so students can formulate and draw a picture illustrating the vertical shift due to increasing or decreasing the amount of area accumulated under the function.

Part V of Tonight's Homework uses the same Multiple Derivatives sheets as students have become very familiar with, but removes the axes for graphing f and f ”.  The purpose here is that students should be able to answer all of the regular multiple derivatives questions based only on graphical features of f ’.