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We won’t be working any examples in this section. This section is here solely for the purpose
of summarizing up all the arc length and surface area problems.
Over the course of the last two chapters the topic of arc
length and surface area has arisen many times and each time we got a new
formula out of the mix. Students often
get a little overwhelmed with all the formulas.
However, there really aren’t as many formulas as it might
seem at first glance. There is exactly
one arc length formula and exactly two surface area formulas. These are,
The problems arise because we have quite a few ds’s that we can use. Again students often have trouble deciding
which one to use. The examples/problems
usually suggest the correct one to use however.
Here is a complete listing of all the ds’s that we’ve seen and when they are used.
Depending on the form of the function we can quickly tell
which ds to use.
There is only one other thing to worry about in terms of the
surface area formula. The ds will introduce a new differential to
the integral. Before integrating make
sure all the variables are in terms of this new differential. For example if we have parametric equations
well use the third ds and then we’ll
need to make sure and substitute for the x
or y depending on which axis we
rotate about to get everything in terms of t.
Likewise, if we have a function in the form 
then we’ll use the second ds and if the rotation is about the y-axis we’ll need to substitute for the x in the integral. On the
other hand if we rotate about the x-axis
we won’t need to do a substitution for the y.
Keep these rules in mind and you’ll always be able to
determine which formula to use and how to correctly do the integral.