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.
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.
To find the volume of the cap of a sphere, we can use the formula for the volume of a spherical cap, which is given by:
V = (1/3)πh^2(3r - h),
where V is the volume of the cap, π is a mathematical constant approximately equal to 3.14159, r is the radius of the sphere, and h is the height of the cap.
In this case, the radius of the sphere is 5 inches and the height of the cap is 1 inch. Let's substitute these values into the formula and calculate the volume.
V = (1/3)π(1^2)(3(5) - 1)
= (1/3)π(1)(15 - 1)
= (1/3)π(14)
≈ (1/3)(3.14159)(14)
≈ 4.18879(14)
≈ 58.9046 cubic inches.
Therefore, the volume of the cap of the sphere is approximately 58.9046 cubic inches.