I don't really understand this question:
Evaluate:
by making a substitution of the form and then interpreting the result in terms of a known area.
I get that I should find a usable then evaluate but don't understand what it means by interpreting in terms of a known area.
Thanks
Okay, I thought I had this sorted out last night but obviously didn't think too much as I worked this problem.
First:
Having gotten
It's not clear to me why we're changing the range from 3 to 9. From a geometric point of view it makes perfect sense but just taking it in the context of the integral I don't see the reasoning.
Second: I thought that it would be simple to evaluate this at first but we have not done anything with squares inside roots. Do I need to make another substitution for this? For instance ?
Thanks for the help so far!
When you change the integral from an integral with respect to x to an integral with respect to u, the terminals need to change to u values.
Since , when and when .
Now, as for solving the integral, like I suggested, you need to make the substitution , and note that when and when and the integral becomes...
which is the same as half of the area of a quarter circle of radius 9 as discussed.
We have yet to introduce a trig function into a integral, or derivative for that matter, that did not already have one in my class. So I never would have even guessed to do this...
Are there rules that precede doing this? Are we able to do this because it is part of a circular function?