"The student had actually gotten his take home test, which had been downgraded to a homework assignment. He worked through it. By the third problem or so, John was doing the problems with no questions! This was a fantastic result. We worked more on the finer points of integrating and tricks that you can use in order to make your work easier, or to have the result look nicer. These are fine tuning points which aren't as important as the I KNOW and the HERE statements described in the previous session. They include bringing your constant out of the integral, using brackets when evaluating the bounds of your integrand, and doing multiple integrations in the same line by writing multiple I know statements on the right hand side. I was extremely proud of him for learning how to do this in two short sessions. He worked through the packet, the first half of which was designated for the evaluation of integrals.
The next main topic that we ended up discussing was how to find the area under a curve or the area between two curves. I told him that the most important thing he should do is to draw a picture of the functions, and then to shade with vertical bars the area under the curve if he is expecting to integrate with respect to the x variable, and to shade with horizontal bars if he expects to integrate with respect to the y variable. The thickness of each of these bars is your dx (if you're integrating in the x direction or your dy if you're integrating with respect to your y direction). Shade one of the bars heavier than the others. Then you have to ask yourself how do I find the area of this bar? The answer is that you multiply the thickness, dx, by the length. However, you have to list the length in terms of the functions. Don't choose real numbers. Once you have this length, place it inside the integral and next step is to figure out what your bounds will be. If one of the bounds is where two functions intersect, then you have to set the two functions equal to each other to solve for x. That will be the bound. Then you can integrate to find the area under the curve.
This brought us to the end of the session. This was a breakthrough session and the student did a fantastic job."