Help with limits of integration

2^Oscar

Hi guys,

I've been doing past paper questions for an exam and I've gotten stuck with the limits of an integral. We have to evaluate

[tex]\int\int\int _{\Omega} \frac{1}{(1+z)^2} dx dy dz[/tex]

where [tex]\Omega = \left\{ (x, y, z) : x^2 + y^2 \leq z^2 \leq 1 - x^2 - y^2, z \geq 0 \right\}[/tex]

using spherical polar coordinates. My problem is finding the limits for r (we use r, theta, phi in lectures), all I get is as far as this inequality [tex] r^2 sin^2 (\theta) \leq r^2 cos^2 (\theta) \leq 1- r^2 sin^2 (\theta) [/tex] and I'm unsure how to go on after this.

I'm sure I'm missing something blindingly obvious, and I'll be fine once I know the limits, but would someone please explain how to proceed and find the limits for r?


Cheers,
Oscar

tiny-tim

Hi Oscar!
the right side would be correct if you used ²

the left side is just y ≤ x

the best way to do this is to ask yourself what shape we're talking about …

try putting [itex]x^2 + y^2 = \rho^2[/itex]

now what is the shape?

2^Oscar

Sorry about that I missed the squares out on the trig functions. I've corrected them to the inequality I actually got!

Wouldn't [itex]\rho^2 = x^2 + y^2 + z^2[/itex] though? I tried rearranging the inequality to use that but couldn't get very far. When I sketched the shape it was [itex] x^2 + y^2 [/itex] coming up and then meeting the same curve coming down from one and forming a kind of egg shape...



Thanks for the reply :)
Oscar

tiny-tim

We usually use r = √(x² + y² + z²) in spherical coordinates, and ρ = √(x² + y²) in cylindrical coordinates.

Try using ρ here, to see what the shape looks like.

2^Oscar

Ahh I see, so in the new shape (a cylinder?) we'd have the limits [itex] \rho \leq z \leq \sqrt{1 - \rho^2}[/itex]?

tiny-tim

No!

Try it with y = 0.