For the Shell method around the y-axis the equation is...

imagine that the part being drilled out is centered on the y-axis. It will have a radius of 5cm. The radius of the sphere will be 12cm.

So the equation of the circle will be

Solving for y and using the positive root we get an equation of the upper half of the circle.

Then using all of this we can set up the integral

Since we used the upper half of the circle we will only get half of the volume so we will need to multiply by 2. Doing so we get..

setting then

subbing in we get...