How do you find the volume of the central part of the unit sphere that is bounded by the planes #x=+-1/5, y=+-1/5 and z=+-1/5#?

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Answer 1 · 2016-07-03T07:39:54

#(2/5) ^3#

#(2/5) ^3#

Answer 2 · 2016-07-03T15:39:30

The volume of the slice between #x=+-1/5# is

= twice the volume of the solid of revolution, about x-axis, of the

area enclosed by the circle

#x^2+y^2=1, x=0, y=0 and x=1/5#

#=2 int pi y^2 d x#, between the limits # x=0 and x= 1/5#

#= 2 pi int (1-x^2) d x#, between the limits

#= 2 pi [x-x^3/3]#, between the limits

#= 2 pi (1/5-1/375)#

#= (148 pi )/375 cubic units.

The three slices for

#x=+-1/5, y=+-1/5 and z=+-1/5#

have this volume and each includes, as intersection, the central

cube bounded by these planes.

So, the required volume #= 3X ((148pi)/375)-2X(2/5)^3#

#=(4/125)(37 pi-4)#

#=3.592# cubic units, nearly..

In making this solid, eight identical wedges, with spherical tops,

have been removed, one from each octant. The volume of each

= (volume of the unit sphere - volume of the solid made)#/8#

#=(4 pi /3-(4/125)(37pi-4))/8#

#= (7pi+6)/375 = 0.0746# cubic units, nearly., cu .