Question

Volume

Answer 1

I'm supposed to find the volume in the space given by \[z \ge \sqrt{x ^{2}+y ^{2}}\]
and \[x ^{2} + y ^{2} + z ^{2} \le 1\]
how do i find the limits of the integration from this??

Answer 2

a sphere of radius 1?

Answer 3

you should find a plotter or just sketch it before you kick off


this is the vol between the unit sphere and the paraboloid

if it were me, i's think you kick off with rectangular and then polarise it


so as a volume integral done in the first octant (thus the x4), it is


\(4 \int\limits_{y = 0}^{1} \int\limits_{x = 0}^{\sqrt{1- y^2}} \int\limits_{z = \sqrt{x^2 + y^2}}^{\sqrt{1 - x^2 - y^2}} dz \ dx \ dy\)

\(= 4 \int\limits_{y = 0}^{1} \int\limits_{x = 0}^{\sqrt{1- y^2}} \sqrt{1 - x^2 - y^2} - \sqrt{x^2 + y^2} \ dx \ dy \)

....and polarise it

Answer 4

@IrishBoy123 how do i figure this out if i don't have a plotter?
and i didn't really the x4 part, could you please explain that?

thank you and sorry for the inconvenience!

Answer 5

is a cone a parabaloid?

Answer 6

nope sir! it'snot :-(

Answer 7

shall i try drawing it if no-one else want to?

Answer 8

i believe its an icecream cone with a scoop :)

Answer 9

so made for spherical!!