calc (please help)

A wedge is cut from a right circular cylinder of radius r by two planes, one perpendicular to the axis of the cylinder and the other making an angle (beta) with the first. Find the volume of the wedge by slicing perpendicular to the y-axis.
Please show all steps

Note: this is the entire question, nothing is missing. This is the problem I got from my math teacher.

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asked by beam
  1. I'm assuming that the angle between the planes is situated on the edge of the cylinder, so that the two planes intersect the cylinder in a circle and an ellipse.

    So, the ellipse has major axis of length 2r secβ, and the far side of the major axis is a distance 2r tanβ below where the planes intersect.

    Due to symmetry, the volume of the wedge is just half the volume of a slice of the cylinder of height 2r tanβ, so

    v = 1/2 πr^2 2r tanβ = πr^3 tanβ

    If you interpret the figure differently, please share your insights.

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    posted by Steve

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