Impossible Spherical Coordinates

Question

This was a question from Indian IIT exam and had 3% solve rate, see if you can solve it within 5 minutes.

Region given by $9 \le x^2+y^2 \le 81$ and $0 \le z \le \sqrt (x^2 + y^2)$

Create a triple integral (volume) by cylindrical coordinates (easy) than attempt to render with spherical.

Is it possible?

Answer 1

In cylindrical:

$$\int_0^{2\pi} \int_3^9 \int_0^r rdzdrd\theta$$

In spherical:

$$\int_0^{2\pi} \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \int_{3\csc\phi}^{9\csc\phi} \rho^2 \sin\phi d\rho d\phi d\theta$$

The second is easy to derive from the first, just use the fact that $r = \rho\sin\phi$ and a small sketch in the $rz$-plane to identify the easiest order of integration.

Answer 2

Yes of course and, with reference to the notation give in the sketch, the limits should be

• $0\le \theta \le 2\pi$

• $0\le \phi \le \frac{\pi}4\quad$

• $\frac 3{\cos \phi}\le r \le \frac 9{\cos \phi}$