TITLE: Applications of Double/Triple Integrals
QUESTION [5 upvotes]: This is the question that I need to solve using mathematica:

The concentration of an air pollutant at a point $(x,y,z)$ is given by: $$p(x,y,z) = x^2y^4z^3 \text{ particles}/m^3$$ We're interested in studying the air quality in a region in 3-space which satisfies $x\ge0$ and is bounded by the $xy$-plane, the surface $z=\sqrt{4-x^2-y^2}$, the plane $y=-2x$, and the $xz$-plane.  Find the total amount of pollutant in this region.

I don't want a full answer to this question, obviously (I need to work it out myself), but I'm just having trouble getting started.  How do I use the bounds here to set up an integral for the problem?  What exactly do the bounds mean/represent in the first place?

REPLY [2 votes]: This problem is conducive to using spherical coordinates.  If you draw a picture, you will see that the effect of the plane $y=-2 x$ is to extend the range of the azimuthal angle to $[0,\pi/2+\arctan{2}]$.  (The azimuthal angle is measured with respect to the positive $x$ axis.)  Therefore, the integral you seek is
$$\int_0^2 dr \, r^2 \int_0^{\pi/2} d\theta \, \sin{\theta} \, \int_0^{\pi/2 + \arctan{2}} d\phi \, p(r \sin{\theta} \cos{\phi}, r \sin{\theta} \sin{\phi},r \cos{\theta}) \\ = \int_0^2 dr \, r^{11} \int_0^{\pi/2} d\theta \, \sin^7{\theta} \cos^3{\theta} \, \int_0^{\pi/2 + \arctan{2}} d\phi \, \cos^2{\phi} \sin^4{\phi}$$
Note that these integrals may be evaluated separately and then multiplied together for the final result.