My Favorite Calculus: Crazy Integrals

While reading on my computer today I received a notification from Dropbox that a shared file called "Crazy Integrals" had been updated.  This was pretty exciting since I created this activity four years ago and even though I am no longer teaching Calculus AB, the teachers are still using this activity.  This also reminded me that I could share this activity for the Explore MTBOS Week 2 blogpost challenge.

Before we launched into our integration chapter I did some pre-work to help students gain a strong understanding of both the notation and the concept of integration.  For Crazy Integrals I started by directing their attention the the diagram below.

 We talked about how finding a definite integral is like finding the area between a curve and the x-axis.  There are some finer details that need to be covered, and we brought them in both in the course of the activity and throughout the chapter.  The first two problems that we talked through are below.  Students noted on their diagram that the top piecewise curve is f(x) and the bottom piece-wise curve is g(x).

Students then discussed the best way to break up each area using vertical lines.  This helped them to rewrite each definite integral as two separate definite integrals.  Then they found each area.  We talked about how in Calculus an area below the x-axis is counted as negative.  Students worked in groups of 3 or 4 to complete the rest of the problems.  

Below are the other diagrams used for the activity.

There were all sorts of great questions that students brought up throughout the activity, most of which called on them to use precise definitions and language to answer.  For example, students wanted to know what to do with the smile and eyes/nose from the clown picture.  We had to use our working definition of definite integral to determine that they were extraneous to the problem.  Students also wanted to know how to find the largest/smallest x-values for the clown's hair.  We had to interpret the associated integrals (below) to know that those parts of the hair were not to be included.  

For the fish diagram students wondered why I only asked them to find the definite integral of the top curve.  Great question, and one that they could probably answer in their own groups.  

Overall super engaging activity with lasting pay-offs in understanding for the remainder of Calculus.  Students didn't want to leave class without finishing this assignment.

Note:  After the first couple of problems students stopped rewriting the definite integral as a sum of definite integrals, and instead got really into showing their work for finding the separate areas.  I figured this was fine as long as they demonstrated the understanding of the skill in the first part of the activity.