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?

  • $\begingroup$ In one octet of the coordinate space,x varies from 0 to 2,y varies from 0 to $\sqrt {4-x^2}$ and z varies from 0 to $\sqrt {9-x^2-y^2}$. Can you take it from here? $\endgroup$ – GTX OC Sep 19 '14 at 11:57

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.

  • $\begingroup$ Thank you!! :). That made a lot of sense :). Managed to get it drawn and solve the rest of the problem with this description :). +1 for your great detail :) $\endgroup$ – user860374 Sep 20 '14 at 9:33
  • $\begingroup$ Thanks - I'm glad this worked out all right for you. (I'll have to learn how to insert pictures someday.) $\endgroup$ – user84413 Sep 20 '14 at 22:07
  • $\begingroup$ You can draw it on any drawing tool (or by hand and take a picture of it), and then upload the picture to a photo sharing site such as imgur or postimage, then copy and paste the link here :) $\endgroup$ – user860374 Sep 20 '14 at 23:13
  • $\begingroup$ Thanks; I will try that out sometime. $\endgroup$ – user84413 Sep 20 '14 at 23:14
  • $\begingroup$ Anytime! :). If you do, please share the sketch you made for this one, incase I may have made a mistake :). $\endgroup$ – user860374 Sep 20 '14 at 23:16

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