# Setting up limits for triple integral in cylindrical coordinates

I want to calculate the volume of this integral by using cylindrical coordinates $$T:Z \le2-x^2-y^2, \ Z^2 \ge x^2+y^2$$ First i want to ask here i don't see the equation for the circle i mean i don't have for example $$x^2+y^2=1 \ or \ x^2+y^2=2x.$$ etc. So this is my first confusion and the second one i am not sure if i am getting the limits right for the limits of integration for Z its obvious i just substitute the equations with the parameters $$x=r\cos\theta, \ y=r\sin \theta$$. To find the limits for r is it right to intersect the two functions ? I am doing so and i get $$r=2-cos\theta-sin\theta$$, so i want to ask if this mean that i must search for where $$(\sin\color{blue}\theta,cos\color{blue}\theta)$$ are negative on the unit circle to get the bounds for $$\color{blue}\theta$$ ? I found the following limits : $$r \le Z \le2-r^2 \\ 0 \le r \le2-\cos\theta-sin\theta \\ \frac{3\pi}{2} \le\theta \le\pi$$

The intersection of the two surfaces is found by solving $$x^2+y^2=2-x^2-y^2$$ which is the circle $$x^2+y^2=1$$

Thus the region of integration is the disk $$0\le r \le 1, 0\le \theta \le 2\pi$$ And the integrand is $$2-2r^2$$

• Thank you. Btw Why we don't integrate just $\iiint d\theta dr dz$ and we use $\iiint 2-2r^2$ this is always confusing me one time we don't use a function to calculate the volume one time we put a function ? – Boris Borovski Jun 18 at 14:08
• The integrand is the distance between the upper and the lower surfaces which in this case is $2-2r^2$ – Mohammad Riazi-Kermani Jun 18 at 14:11
• Thanks. It totally makes sense now. – Boris Borovski Jun 18 at 14:14
• Thanks for your attention. – Mohammad Riazi-Kermani Jun 18 at 14:15

HINT

Note that you have $$x^2+y^2 \le z \le 2 - x^2-y^2,$$ and the projection of this into the $$xy$$-plane occurs where the boundaries intersect, i.e. when $$x^2+y^2 = 2 - x^2 - y^2 \iff x^2+y^2=1,$$ so we will have to integrate this over the disk $$D$$ of radius $$1$$.

Can you now set up the integration?