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EXAMPLE 4Finding the Volume of a Solid Using Spherical Coordinates

Find the volume of the solid that is removed from a hemisphere of radius 1 when it is cut by a cone that makes an angle of 30 with the positive z-axis.

SolutionFigure 61 shows the part of the hemisphere cut by the cone, namely, the volume under the sphere and inside the cone. The upper surface is the hemisphere of radius r=1; the lower surface is the xy-plane (z=0). The angle θ ranges from 0 to 2π and ϕ ranges from 0 to π6. 0ρ10θ2π0ϕπ6

Then the volume V of the solid is V=