A cap of a sphere is generated by rotating the region about the y-axis. Determine the volume of this cap when the radius of the sphere is 5 inches and the height of the cap is 1 inch.

Answer 1

The cap outside is determined
x^2+y^2=25

YOu are interested in the region from y=4 to y=5

So integrating some xdy from 4 to5 will do it, and rotating it (2PI degrees)

volume=PI r^2 dy where r is x outer limit, or
volume= INT PI (25-y^2) dy from 4 to 5

= PI [25y -y^3/3) from 4 to 5
= PI (125-125/3-100+64/3)

check that.