What is volume of the cap of the sphere with height H?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to consider the region as a part of a circle bounded by axes x=0 and y=0 and you need to evaluate the volume of solid of revolution resulted if you rotates the circle around y axis such that:

`V = int_(R-H)^R pi*R^2 dy`

You need to remember the standard equation of  circle such that:

`(x - h)^2 + (y - k)^2 = r^2`

Considering h=0 and k=0 yields:

`x^2 + y^2 = R^2 =gt x = sqrt(R^2 - y^2)`

`V = int_(R-H)^R pi*(R^2 - y^2) dy`

`V = pi(R^2y - y^3/3) |_(R-H)^R`

`V = pi(R^3 - R^3/3 - (R-H)R^2 + (R-H)^3/3)`

`V = pi(R^3 - R^3/3 - R^3+ R^2H + R^3/3 - R^2H + RH^2 - H^3/3)`

Reducing like terms yields:

`V = pi(RH^2 - H^3/3)`

Hence, evaluating the volume of the cap of the sphere under given conditions yields  `V = pi(RH^2 - H^3/3).`