Calculus III - Conceptual Help

[Beautiful]

I'm taking a Calc III Course and I want to know if anyone out there can help me with a few conceptual ideas. I know how to do the math but I am missing the conceptual idea of it.

Why should/do we we parametrize?
Why should/do we convert to Polar or Cylindrical?

These basic ideas should be easy enough for me at this point but for some reason I keep mixing things up. I tried to searching the web and other sources but I keep finding information on the actual process, and they all pass the big idea I am looking for. Idk maybe I'm not the best at searching these things...

Thanks in advance. I really appreciate it

 

[SteamKing]

We do these things (parameterization, changing coordinate systems) to make analyzing a problem easier.

To take a simple example, we can calculate the area of a circle using cartesian coordinates and some sophisticated integration and come up with A = πr$^{2}$. By changing the problem to polar coordinates,

A = $\int$$^{2π}_{0}$ r$^{2}$/2 dθ,

where r = radius of the circle = constant, which is much easier to evaluate than

A = 2*$\int$$^{r}_{-r}$ (r$^{2}$-x$^{2}$)$^{1/2}$ dx

 

[Beautiful]

Understood, I need more practice.

I want to be able to quickly identify which coordinate system to convert to when given a problem. It doesn't seem like there is any sort of general case because problems can be done in multiple ways. Am I accurate with that assumption?

 

[SteamKing]

Yes, you are. This is why some things about studying math and science only come with the experience of working out a variety of problems.