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.

To find the volume of the wedge, we will use the method of slicing perpendicular to the y-axis.

Step 1: Visualize the problem
First, let's visualize the situation. Imagine a right circular cylinder with a radius of r. Two planes cut the cylinder: one perpendicular to the axis of the cylinder, and the other making an angle β with the first plane. The resulting shape is a wedge.

Step 2: Identify the variables
In this problem, we have the following variables:
- r: the radius of the cylinder
- β: the angle between the two cutting planes

Step 3: Determine the limits of integration
Since we are slicing perpendicular to the y-axis, our limits of integration will be based on the y-coordinate of the cylinder.

Let's call the height of the cylinder H. The limits of integration will be from 0 to H, as we consider each slice of the wedge from the bottom to the top.

Step 4: Express the volume of each slice
To find the volume of each slice, we can consider an infinitesimally thin slice at height y. The thickness of the slice will be dy, and its width will be 2r at the bottom and 2r - 2y tan(β) at the top.

Therefore, the volume of each slice can be expressed as dV = π(2r - 2y tan(β))^2 dy.

Step 5: Integrate to find the total volume
To find the total volume, we integrate the expression for each slice over the limits of integration.

V = ∫[0 to H] π(2r - 2y tan(β))^2 dy

Step 6: Simplify and solve the integral
Expand the expression inside the integral, and then integrate term by term.

V = ∫[0 to H] π(4r^2 - 8r y tan(β) + 4y^2 tan^2(β)) dy

V = π ∫[0 to H] (4r^2 - 8r y tan(β) + 4y^2 tan^2(β)) dy

V = π [4r^2 y - 4r y^2 tan(β) + y^3 tan^2(β)/3] evaluated from 0 to H

V = π [4r^2 H - 4r H^2 tan(β) + H^3 tan^2(β)/3]

Step 7: Simplify the expression
Now, we can simplify the expression for the volume by factoring out common terms.

V = π [H (4r^2 - 4r H tan(β) + H^2 tan^2(β)/3)]

V = π H [4r^2 - 4r H tan(β) + H^2 tan^2(β)/3]

And there we have the volume of the wedge, expressed in terms of the given variables r and β, as well as the height H of the cylinder.

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.