Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.
xy = 2, x = 0, y = 2, y = 4
To find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis using the method of cylindrical shells, we need to set up an integral expression.
First, let's visualize the region bounded by the curves: xy = 2, x = 0, y = 2, and y = 4. This region is a rectangular area in the first quadrant, with the curve xy = 2 as its boundary.
Since we are rotating this region about the x-axis, each cylindrical shell will be a thin strip with width Δx along the x-axis. To find the volume of a single cylindrical shell, we need to find its height and circumference.
Consider a small strip at a distance x from the y-axis. The width of this strip is Δx, and its height is given by the difference between the upper and lower curves: y = 4 - y = 2 = 2.
The circumference of the shell is equal to the circumference of a circle with radius x. So, the circumference is given by 2πx.
The volume of the cylindrical shell is then given by the product of its height, circumference, and width: ΔV = 2πx * 2 * Δx.
To find the total volume, we need to sum up the volumes of all the cylindrical shells. We can do this by setting up an integral expression:
V = ∫[from x = 0 to x = 2] 2πx * 2 dx.
Integrating this expression will give us the volume of the solid obtained by rotating the region about the x-axis.
To find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis using the method of cylindrical shells, we need to integrate the volume of each individual shell.
The region bounded by the given curves is a rectangle in the xy-plane with vertices (0, 2), (0, 4), (2, 4), and (2, 2). When rotated about the x-axis, this rectangle will generate cylindrical shells.
Let's calculate the volume of a single cylindrical shell first. Consider a vertical strip within the rectangle bounded by x and x + Δx. The height of this strip is given by the difference between the y-values of the curves at x.
The radius of this cylindrical shell is the x-value, and the height is the difference between the y-values of the curves at x. Therefore, the volume of this shell can be calculated as follows:
dV = 2πxΔx * (y_upper - y_lower)
To find the total volume, we need to integrate this expression from x = 0 to x = 2:
V = ∫[0, 2] (2πx * (y_upper - y_lower)) dx
We can express the equation of the curves as y = 2/x and y = 4. Substituting these values into the integral expression, we get:
V = ∫[0, 2] (2πx * (4 - (2/x))) dx
Simplifying further:
V = 2π ∫[0, 2] (4x - 2) dx
V = 2π ∫[0, 2] (4x) - ∫[0, 2] (2) dx
V = 2π * [2x^2] from 0 to 2 - 2 * [x] from 0 to 2
V = 2π * (2(2^2) - 2(0^2)) - 2 * (2 - 0)
V = 2π * (8 - 0) - 4
V = 16π - 4
Therefore, the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis is 16π - 4 cubic units.
Each shell has thickness dy and height x = 2/y
So, the volume v is
v = ∫[2,4] 2πrh = ∫[2,4] 2πy(2/y) dy
= ∫[2,4] 4π dy
= 8π
As a check, you can use discs, getting
v = π(4^2-2^2)(1/2) + ∫[1/2,1] π(R^2-r^2) dx
where R=y and r=2
v = 6π + ∫[1/2,1] π((2/x)^2-2^2) dx
v = 6π+2π = 8π