First off, I'm having a bit of trouble changing this integral's bounds so that I can evaluate it starting by integrating the dx part first. Any helps/tips will be appreciated.
Second, I've just finished learning how to change variables into polar coordinates, but I'm still a bit confused on how to go about doing so. Here is an example I'm trying to evaluate, but I'm not quite sure where to start:
While drawing the region of integration is the easiest,safest way out there, sometimes using inequalities will work too. If you have more algebraical affinities compared to geometrical, you can manipulate the inequalities.
The first bound says
The second bound says
Remember that both and are increasing functions.
Raise all sides to the power e.
So clearly and
So the bounds on x are e^y and 3, while the bounds on y are 0 and ln 3
Thanks everyone! mr fantastic, for the first one, I had those exact bounds when I was trying it (I drew it out as well), but I guess I was making an error somewhere midway in the rest of my calculations (seems to work just fine on this try).
Isomorphism, that is an awesome way of figuring out the bounds! This will definitely help with my upcoming assignment. Thank you for sharing that!