



 
Hi Rory, I started with the circle in the plane with centre at the origin and radius r. I then sliced the circle with the line y = h, h > 0. (I am using h where you used x in your expression.) I then took the region inside the circle, above the line y = h and in the first quadrant, and revolved it about the yaxis to get a cap of the sphere.
I then used the method of cylindrical shells to find the volume of this cap and called the volume c(h). The volume of the remainder of the sphere is 4/3 r^{3}  c(h). These volumes are to be in the ratio of m to n where n > m so
I then simplified this expression and compared it to the expression you sent us, with x replaced by h. I hope this helps,  


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