# Choosing the best coordinate system for triple integrals?

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.

• So what was your reasoning? I am looking for some intuition to apply in general cases.
– abe
Dec 12, 2015 at 3:18
• Also, are you sure your boundaries are correct?
– abe
Dec 12, 2015 at 3:23
• Thanks for reminding about boundaries. I fixed it. Dec 12, 2015 at 3:27
• Thanks for answering and providing some back log! I know we can solve in any coordinate system, good point, and as you said, some are simpler than others. I will keep an eye out for the similar terms.
– abe
Dec 12, 2015 at 3:31