# Volume under a Sphere Tangent to a Cone

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The red circle with unit radius and center on the positive axis is tangent to the lines through the origin with slopes .

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Contributed by: Abraham Gadalla (May 2012)

Open content licensed under CC BY-NC-SA

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If the points of tangency are and , where is the slope, then the radius of the tangent circle at these points can be calculated to be , with corresponding tangent circle center at .

The volume under the sphere is equal to , where:

1. is the total volume resulting from rotating the region bounded by the line connecting the center of the circle with the point of tangency about the axis. This total volume is calculated using the shell method; .

2. is the volume of the spherical sector.

The volume of a sector of a sphere with radius is , where is the height of the segment. In this case, . Therefore ,

so .

Finally, the required volume is .

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