Question

Let D be the smaller cap cut from a solid ball of radius units by a plane from the center of the sphere. Express the volume of D as an iterated triple integral in ​spherical, ​cylindrical, and rectangular coordinates. Then find the volume by evaluating one of the three triple integrals.

Answer 1

we get:

= 1/2 ʃ02π ʃ14 ( √u -1)du dθ

= ʃ02π 11/12 dθ

= 11/6 π

Explanation:

Given:

Radius = 2 units

Plane 1 unit from center of sphere.

The volume of D as an triple integral in spherical, cylindrical and rectangular coordinates are:

Spherical:

ʃ02π ʃ0π/3 ʃsecΦ2 p2 sinΦ dp dΦ dθ

Cylindrical:

ʃ02π ʃ0√3 ʃ1√4-r2 r dz dr dθ

Rectangular:

ʃ-√3√3 ʃ-√3-x2√3-x2 ʃ1√4-x2-y2 1dz dy dx

Solving the integral by using cylindrical coordinates:

ʃ02π ʃ0√3 ʃ1√4-r2 r dz dr dθ = ʃ02π ʃ0√3 r ( √(4-r2) -1) dr dθ

put u = 4-r2, by substituting,

we get:

= 1/2 ʃ02π ʃ14 ( √u -1)du dθ

= ʃ02π 11/12 dθ

= 11/6 π