Question

volume of solid bounded by cone x^2+y^2=z^2 and paraboloid x^2 +y^2=z above xy plane....ty
picture attached at second post....ty

Answer 1

I did first to try to find intersection of both cone and paraboloid is it?

Answer 2

@ash2326 this is the final question for now i need your help hope you dun mind? ty so much for the other 4....

Answer 3

angalc557 I'm so sorry but I don't know this one. Ask for help in chat.

Answer 4

\[z^2=x^2+y^2\quad,\quad z=x^2+y^2\]\[z^2=z\quad\Rightarrow\quad z=0\quad\vee\quad z=1\]\[S:\quad x^2+y^2=1\]\[V=\iint\limits_S\left(\sqrt{x^2+y^2}-(x^2+y^2)\right)dxdy\]\[x=\rho\cos\phi\quad,\quad y=\rho\sin\phi\quad,\quad dxdy=\rho d\rho d\phi\]\[V=\iint\limits_S\left(\rho-\rho^2\right)\rho d\rho d\phi=\int\limits_0^{2\pi} d\phi\int\limits_0^1 \left(\rho^2-\rho^3\right)d\rho=\]\[=2\pi\left(\frac{\rho^3}{3}-\frac{\rho^4}{4}\right)|_0^1=2\pi\left(\frac{1}{3}-\frac{1}{4}\right)=\frac{\pi}{6}\]

Answer 5

@nikvist nicely guided hmm i would like to ask the cone is facing downwards and the paraboloid is facing upwards in terms of 3d maybe you could just sketch it out the image for me to clearly understand....ty ;)

Answer 6

2D space

Answer 7

so if written in words means the region of integration is bounded from above or below the paraboloid and from below or above the cone.....hard to imagine haha....? ty

Answer 8

cone-above , paraboloid-below

Answer 9

hmm @nikvist actually i was thinking few days i thought the volume of D is between the outer parabola and the inner conic can u refer my drawing to confirm....haha right? or my drawing is wrong...??

Answer 10

picture is OK

Answer 11

means in that case the region bounded is the paraboloid minus the cone but ur equation is cone minus paraboloid right?

Answer 12

cone-paraboloid

Answer 13

means the region is between both lines or inside the conic area?

Answer 14

so i thought is the outer minus the inner volume so means paraboloid minus cone could u enlighten me why u think is cone minus paraboloid based on the diagram...?? i so confused....ty so much :)

Answer 15

between both lines

Answer 16

i see that what i was imagining so means i thought the volume normally will be the outer minus the inner right? somewhere i am wrong?

Answer 17

your method is OK, but you can't find volume of parabolid without integration.

Answer 18

let me just ask for my extra knowledge @nikvist why cant be volume of paraboloid minus the cone in the volume equation u stated....?? thanks