I am doing my homework and I got to one exercise where I have to write the integral for this set:
$$B=\{(x,y,z)\in \mathbb{R}^3 : \sqrt{x^2+y^2}<z<2-\sqrt{x^2+y^2}; y>0\}$$
using cylindrical coordinates.
The problem is that I am having a hard time to interpret this set. I tried to draw the region but I guess I did not draw it the right way.
Can someone explain me how to interpret this particular set?
Thanks!
Both of the bounds are shaped by a cone, one going up, and the other going down. Where they intersect is also interesting; setting one equal to the other, you get $$2\sqrt{x^2+y^2}=2$$ or the unit circle. Given you have the bounds on $z$ laid out, and the planar bounds are within the unit circle, the integral looks like $$\int_0^{2\pi} \int_0^1\int_{r}^{2-r}rdzdrd\theta$$