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.

we get:      

                                        = 1/2  ʃ02π  ʃ14    ( √u -1)du dθ

                                        = ʃ02π   11/12 dθ

                                        = 11/6 π

Step-by-step 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