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Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume of the smaller wedge...

Use cylindrical or spherical coordinates, whichever seems more appropriate. Find the volume of the smaller wedge cut from a sphere of radius 2 by two planes that intersect along a diameter at an angle of ?/2.

Solutions

Expert Solution

we can see that the givenwedge is described by the following inequalities in spherical coordinates:




Hence the volume is given by the integral

here Z means intergral sign-

If we had let range from 0 all the way to , we would have obtained the volume of the full sphere, which is . This observation leads to a quick way to check our answer geometrically. Since ?/2 is one fourth of , we should expect the wedge to have one
fourth of the volume of the sphere. Indeed,

(shortcut)

genius_generous answered 2 years ago
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