Find the volume of the part of the sphere which is inside the cylinder where are the cylindrical coordinates.
My attempt : I'm not able to visualize the sphere nor the cylinder.
For the sphere, does ? If so then I can imagine the sphere. (Centered at the origin and with radius ).
For the cylinder I don't know how to go further. I also feel it's useless to convert all in Cartesian coordinates but this way it's easier to me to visualize the volumes.
So the sphere is centered at the origin and has a radius of .
The cylinder is centered in and has a radius of and is parallel to the z-axis.
I now can visualize the volume. However I don't know how to set up the triple integrals. Should I use Cartesian coordinates? Must I find the ellipses of intersection between the sphere and the cylinder? How would you proceed?
is simply a sphere of radius centered at the origin.
following Mr F's lead, the cylinder is
in the xy-plane, the trace of this cylinder is a circle with radius a/2 centered at (0, a/2)
now, you are bounded above and below by the sphere. we can simply find the volume in the first quadrant, and then multiply the answer by 4.
hence, we integrate the function 1, with the following limits:
, and
and don't forget to multiply by 4
any questions?
that's not the integral you want to do first. you leave the one with the constant limits for the last. anyway, an integral like that would conventionally be done using trig substitution
in the equation for the cylinder, solve for xAnd I'm having some troubles to realize that goes from to
solve for z in the equation of the sphere.and from to
then note that you only care about the first quadrant.
Yes I know, it was the second integral, the first being the one with dz.
ok thanks. I think I won't forget this in future.in the equation for the cylinder, solve for x
solve for z in the equation of the sphere.
yes I realized thanks to your explanations.then note that you only care about the first quadrant.
In cylindrical coordinates, I have a doubt about the second integral (dr) : . (Or is it . ?)