## Help with limits of integration

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

$$\int\int\int _{\Omega} \frac{1}{(1+z)^2} dx dy dz$$

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

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 $$r^2 sin^2 (\theta) \leq r^2 cos^2 (\theta) \leq 1- r^2 sin^2 (\theta)$$ 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

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Hi Oscar!
 Quote by 2^Oscar … all I get is as far as this inequality $$r^2 sin(\theta) \leq r^2 cos(\theta) \leq 1- r^2 sin(\theta)$$
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 $x^2 + y^2 = \rho^2$

now what is the shape?

 Sorry about that I missed the squares out on the trig functions. I've corrected them to the inequality I actually got! Wouldn't $\rho^2 = x^2 + y^2 + z^2$ though? I tried rearranging the inequality to use that but couldn't get very far. When I sketched the shape it was $x^2 + y^2$ coming up and then meeting the same curve coming down from one and forming a kind of egg shape... Thanks for the reply :) Oscar

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## Help with limits of integration

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.

 Quote by 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.
Ahh I see, so in the new shape (a cylinder?) we'd have the limits $\rho \leq z \leq \sqrt{1 - \rho^2}$?

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