I am working with double and triple integral in multi-variable calculus and have found that it is extremely useful to convert between different coordinate systems including:
Spherical: Cylindrical: Polar: and Cartesian Well, I know the conversion values and how to transition variables over.
- My question is, what should I look out for when determining which one to convert to?Example:
$$\iiint (3y^2+3x^2) dzdydx$$
^^^triple integral bounded by the paraboloid $z=x^2+y^2$ and the plane $z=25$
Finding the bounds is easy, but converting to a useful coordinate system is the challenge.
Go for Cylindrical
boundary:
$$z=r^2,~z=25$$
integration:
$$\iiint 3r^2 \times rdr d\theta dz$$
$$\int_{r=0}^{5}\int_{z=r^2}^{25}\int_{\theta=0}^{2\pi} 3r^2 \times rdr d\theta dz$$
There is no reason. All coordinations work. It is matter of comfort. I chose cylindrical because I see $(x^2+y^2)$ both inside the integral and in the boundaries. It reminds $r$ from cylindrical coordination to me. Also $z$ acts separately from $r$. So Spherical brings trouble. Cartesian bring so many unwanted root squares for the boundaries of individual integrals.