I'd like to ask you for help with these three problems I found.
- Evaluate the volume of the solid region $R = \{(x,y,z): x^2 + y^2 \le z \le 4 - \sqrt{x^2+y^2}\}.$
- Evaluate the volume of the solid that lies below the paraboloid $z = x^2+y^2$ and inside the sphere $x^2 + y^2 + (z-2)^2 = 9.$
- Evaluate the upward flux of the vector field $F(x,y,z) = (x+y)\mathbf{i} + (y-z)\mathbf{j} + z^2\mathbf{k}$ through the part of the paraboloid $z = x^2 + y^2$ cut off by the plane $z = 2.$
My main problem is that i can't figure out regions of integration. I'd really appreciate if someone would show me some way to find the region so I can get the scheme and be able to use it in next problems. Thank you in advance!
I would use cylindrical coordinates in all cases.
For 1, you get $$r^2\le z\le 4-r$$ Just focus on the $r$ terms. This will give you the integration range for $r$ as the interval between $\frac{-1\pm\sqrt {17}}{2}$, so you can find then $z$.
For 2, the lowest $z$ is $0$ (bottom of the paraboloid), and the top is $z=5$ (top of the sphere). Note that you have two integration regions, depending if the radius of the paraboloid, or the radius of the sphere is larger. By radius I mean the radius in cylindrical coordinates. The intersection is given by $$x^2+y^2=r^2=z=9-(z-2)^2$$ If I did not make any mistake, the positive solution is $\frac{3+\sqrt{29}}{2}$
For 3, you have a disk of radius $\sqrt 2$ in the plane $z=2$