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
Is the intersection of the planes tangent to the curved edge of the cylinder?
Which axis is the axis of the cylinder? the y-axis? If so, why mention it by name?
I don't know, this is the problem I got from my math teacher
To find the volume of the wedge, we need to break it down into smaller slices and then integrate.
Let's start by visualizing the situation. We have a right circular cylinder, and we're cutting it with two planes. One plane is perpendicular to the axis of the cylinder, and the other plane makes an angle β with the first plane.
We will slice the cylinder perpendicular to the y-axis. Let's consider a small slice, dy, at a distance y from the y-axis.
The width of this slice will be the circumference of the cylinder at that height, which is 2πr. The length of this slice will be dy.
The height of the slice is given by the equation of the line that represents the second plane. Let's write this equation in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept. Since the plane makes an angle β with the first plane, the slope will be tan(β). So, the equation becomes y = tan(β)x + b.
Now, let's find the y-intercept. At the y-axis, x = 0. Plugging this into the equation, we have 0 = tan(β)(0) + b, which simplifies to b = 0.
So, the equation becomes y = tan(β)x.
To find the volume of the wedge, we need to integrate the volume of each slice from 0 to the height of the cylinder, h.
The volume of each slice will be the width times the length times the height:
dV = 2πr * dy * (tan(β)x)
Integrating from 0 to h, we have:
V = ∫[0 to h] (2πr * dy * (tan(β)x))
Let's integrate with respect to y:
V = 2πr * ∫[0 to h] (tan(β)x)dy
Since x is a function of y (x = y/tan(β)), we can substitute it in:
V = 2πr * ∫[0 to h] (y/tan(β))dy
Simplifying:
V = (2πr/tan(β)) * ∫[0 to h] y dy
V = (2πr/tan(β)) * [y^2/2] from 0 to h
Evaluating the definite integral:
V = (2πr/tan(β)) * [(h^2 - 0^2)/2]
V = πr^2h/tan(β)
Therefore, the volume of the wedge sliced perpendicular to the y-axis is given by V = πr^2h/tan(β).
To find the volume of the wedge, we can use integration.
Step 1: Setting Up the Problem
Let's visualize the wedge by considering a coordinate system. Assume that the cylinder's axis lies along the x-axis. The plane perpendicular to the axis will cut the cylinder in a circle of radius r. The second plane will intersect this circle, creating a wedge-shaped region.
Step 2: Finding the Equations of the Planes
The equation of the plane perpendicular to the axis is x = 0. This plane cuts the cylinder at the origin (0,0,0) and the circle x^2 + y^2 = r^2.
The equation of the second plane can be written in slope-intercept form as y = mx, where m = tan(beta). This plane intersects the cylinder along a line on the circle.
Step 3: Finding the Intersection Points of the Planes with the Circle
To find the intersection points, we substitute y = mx into the equation of the circle:
x^2 + (mx)^2 = r^2
Simplifying, we get:
(1 + m^2) x^2 = r^2
x^2 = r^2 / (1 + m^2)
Taking the square root, we have:
x = ± (r / sqrt(1 + m^2))
Since the wedge is symmetric, we only need to consider the positive x-values.
Step 4: Setting Up the Integral for Volume
To find the volume of the wedge, we will slice it perpendicularly to the y-axis. Consider a small slice at a y-value between -r and r. The width of this slice is dy, and the length of the slice varies with y.
To find the length of the slice, we can use the equation of the second plane: y = mx. Rearranging, we have: x = y/m
The total volume of the wedge is then given by integrating the volume of each slice from y = -r to y = r.
Step 5: Integrating to Find the Volume
The volume of a small slice is given by the cross-sectional area multiplied by the width dy:
dV = (pi * (x^2) * dy)
Substituting for x in terms of y:
dV = (pi * ((y/m)^2) * dy)
Integrating from y = -r to y = r:
V = ∫[from -r to r] (pi * ((y/m)^2)) dy
Simplifying, we have:
V = pi / m^2 * ∫[from -r to r] (y^2) dy
Evaluating the integral, we get:
V = pi / m^2 * (y^3 / 3) |[from -r to r]
Substituting the limits:
V = pi / m^2 * (r^3 / 3 - (-r)^3 / 3)
V = pi / 3m^2 * (2r^3)
Step 6: Finalizing the Solution
The volume of the wedge by slicing perpendicular to the y-axis is given by:
V = (2pi/3) * (r^3 / m^2)
Therefore, the volume of the wedge is (2pi/3) * (r^3 / m^2).