I am given the following question
Let $D$ be the region in $\mathbb{R}^3$ that lies within $x^2 + y^2 =4$, underneath the surface $z= 4- x^2 - y^2$ and above the surface $z=- \sqrt{9-x^2 - y^2}$
1) Draw $D$ in $\mathbb{R}^3$ and, on the sketch, indicate the projection of $D$ onto the $XY$-plane. Name this region $E$.
2) Find the volume of $D$ using triple integrals (in cylindrical coordinates). $\textbf{(I will be able to do this part once I have Question 1)}$
Can someone please assist me in how to go about sketching this region? Or provide me with a Mathematica (or similar) sketch of the region?
Since I don't know how to provide sketches in this environment, I will try to give a verbal description of D and E:
D essentially consists of 3 pieces:
1) Its top is the portion of the paraboloid $z=4-x^2-y^2$ which lies above the xy-plane. (Notice that the intersection of the paraboloid with the xy-plane is the circle $x^2+y^2=4$.)
2) Its bottom is the portion of the hemisphere $z=-\sqrt{9-x^2-y^2}$ which lies within the cylinder $\;\;\;\;x^2+y^2=4$. (This hemisphere is the bottom half of the sphere $x^2+y^2+z^2=9$.)
3) Its side is the portion of the cylinder $x^2+y^2=4$ extending down from the xy-plane to its intersection with the hemisphere.
Since the region D lies within the cylinder $x^2+y^2=4$ and includes the region in the xy-plane enclosed by the circle $x^2+y^2=4$, E is just this circular region.