Mastering Mathematics Smartly
by Wee Wen Shih

How do you find the area under a curve that has been given parametrically? 

Let this document that contains a short summary together with an accompanying worked example show you the way.

  Students often find it difficult to sketch with GC curves that are defined parametrically.

  The typical issues for students using TI-GC are in deciding the values for Tmin, Tmax and Tstep.

  Let's consider the following example for our discussion:

  The curve C is defined parametrically by

  x = (1 + t)^(2/3), y = ln (t^2), t <= -1.

  Find the area of the region enclosed by C, the lines x = 0 and x = 1, and the x-axis.

  First, we need to sketch the curve and identify the region visually before we can arrive at a definite integral.

  The lines x = 0 and x = 1 give us some indication of what values to set for Tmin and Tmax.

  When x = 0, t = -1 obviously.

  When x = 1, (1 + t)^(2/3) = 1 from which t = -2 is obtained. We need to reject the other possibility t = 0, as we are given that t <= -1 and we know that y = ln (t^2) is not well-defined when t = 0.

  So, we set Tmin = -2 and Tmax = -1.

  Never assume that Tmin and Tmax are going to take positive values, as many students tend to do so for the sake of simplicity. School exams often expect approaches that run contrary to students' thinking :P

  What is Tstep, you may ask? It's the interval between two consecutive T values. Say, we set Tstep to be 0.1, then we'll have the GC to plot the curve for these values:

  -2 (Tmin), -1.9, -1.8, -1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1 (Tmax).

  We have to be careful to select a reasonable Tstep, or we'll risk having an inaccurate graph. For example, if we were to let Tstep be 0.5, then the GC will plot the curve inaccurately for these values:

  -2 (Tmin), -1.5, -1 (Tmax).

  You may try and compare both graphs on your GC to see what I mean. The bottomline is more points, better accuracy.

  If one were to apply a Tstep value of 0.01, he/she will lose precious time because the GC will take a while to generate the curve over many values of T:

  -2 (Tmin), -1.99, -1.98, ..., -1.03, -1.02, -1.01, -1 (Tmax).

  Do try this on your GC and appreciate its slowness. So, another bottomline is more points, but more to a resonable extent.