The volume of a cap of a sphere with radius r and height h.
Definition of volume:
Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is
Radius of sphere is r and height of cap is h.
Consider a cross section of the cap at height h from the origin as shown in above figure.
To find the cross section,
Consider a circle x2+y2=r2
Subtract y2 from both sides,
Taking square root to find the value of x,
From the figure, limits of integral are from r-hto r.
So, the volume of the cap of sphere is
By substituting the limits,
The volume of a cap of a sphere with radius r and height h is πh2r-h3