Find the area of the portion of the sphere of radius 2 (centered at the origin) that is in the cone
What I have so far is that:
=
=
Then I have But is wrong, what is wrong?
Thank you.
As Danny says, you should write the equation of the sphere as . To find the region over which to integrate, you need to know where the sphere meets the cone, namely where or in other words . So the integral will be over the region , and the surface area is given by the integral . This cries out for conversion to polar coordinates, and it then becomes . From there on, it's easy.
What about if the sphere were radius 5? I seem to be goofing it up just a little somewhere when converting to polar/spherical coordinates. Also, it seems as though the Jacobian should be (p^2)sin(phi) instead of just r because this should be changed to polar and not cylindrical coordinates... right? Any feedback, insight or explanation of this integral problem would be fantastic as I am seriously stuck and puzzled by this problem. Thanks.
I think it would be easier to do the all thing in spherical coordinates.
Notice that if the height of the cone is z (a generic point z), and the radius of the base is , so the generating angle for the cone, , is given by
as , and
this makes the integral for the area
it seems much more simpler to me, and no need to change the coordinates in between. the R being 2 or 5 makes no difference you just replace it in the integral as a constant.