Screencasting some calculus

OK, no CAS here, but the power of screencasting allows one to add a voice-over to problems that are difficult to capture in the linear text that predominates Web-based writing.  The following two problems were requests from March 6-7, 2012 on the AP Calculus EDG.  My solutions are screencasts using the ShowMe Whiteboard app on my iPad.  Thanks to my former student (now technology consultant) Beth Holland for the lead to that great app.

  1. A region is defined by y=e^x, x=1, x=8, and the x-axis.  What is the volume of the solid that results when that region is revolved around the line y=4?  Click here for my solution.

As an aside, colleague John Burk (@occam98) pointed me to a cool article, Volumes of Revolution & 3D Modeling, that quite rightly identifies the most challenging part about volumes of revolutions for students:  visualization.  This is another reason why we need to make greater use of screencasting or other technologies to get past the walls of words.

  1. A curve is defined parametrically by x=t^2+1 and y=4t^3+3.  What is the length of that curve for -1\le t \le 0?  Click here for my solution.

I find that many students (and some teachers) have difficulty sorting out the different forms of arclength.  My screencast describes very briefly why I believe students should work from the single arclength formula, \int \sqrt{dx^2+dy^2}.

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