I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just started again with the basics, cartesian to polar and already came to a problem. if $r=4\cos(\theta)$ how do I know that its centered at $(2,0)$ and has a radius of $4$ from lets say a plug and chug into the formula method. I'd prefer to just substitute the relevant things into the transformation equations, I know that seems a bit lazy because Im not analyzing the maths properly, but for now I just want to have sure limits, a method that will always give me what I need. What I really am asking for is how to I convert to these other coordinate systems and find the limits.
2026-03-29 14:17:21.1774793841
Converting coordinates
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I don't know if the answer addresses your case, but due to no answers, here is mine. The standard formulae, which connect the cartesian and the polar coordinates are: $$ x = r\cos\theta \text{ and } y = r\sin\theta.$$
Setting $r = 4\cos\theta$, we get: $$x = 4\cos\theta \cos \theta = 4\cos^2 \theta.$$ Similarly, we get: $$y = 4\cos\theta \sin\theta.$$
Thus: $$x^2 + y^2 = 16\cos^4\theta + 16\cos^2\theta\sin^2\theta = 16\cos^2\theta(\cos^2\theta + \sin^2\theta) = 16\cos^2\theta=4x.$$
Thus, we have the equation: $$\begin{array}[t]{l}x^2 + y^2 - 4x=0\\ x^2-4x+4+y^2 = 4\\ (x-2)^2+y^2=4, \end{array}$$ which is the equation that describes a circle centered at $(2,0)$ with radius $\rho=2.$