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?
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 don't forget to multiply by 4
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.
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 . ?)