0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.