The volume of the solid obtained by rotating the region enclosed by

y=x^2 y=2x

about the line x=2 can be computed using the method of disks or washers via an integral

The volume is V

To find the volume of the solid obtained by rotating the region enclosed by the curves y = x^2 and y = 2x about the line x = 2, we can use the method of disks or washers.

Let's start by visualizing the region of interest. The curves y = x^2 and y = 2x intersect at two points: (0, 0) and (2, 4). So the region is bound by the x-axis, the line x = 2, and the two curves.

To use the method of disks or washers, we need to consider a small element within the region (a disk or a washer). We will rotate this element about the line x = 2 to form a small volume. By summing up these small volumes, we can obtain the total volume.

Let's consider a vertical strip of width dx at an arbitrary x-value within the region. This strip will have a length difference of y = 2x - x^2 between the two curves. Rotating this strip will form a disk or washer.

For the method of disks, the volume of each disk is given by V_disk = π * r^2 * dx, where r is the radius of the disk. In this case, the radius is the distance between the x-value and the line x = 2.

For the method of washers, the volume of each washer is obtained by subtracting the smaller disk's volume from the larger disk's volume. The radius of the larger disk is still the distance between the x-value and the line x = 2, while the radius of the smaller disk is the distance between the x-value and the curve y = x^2.

To find the bounds for integration, we need to determine the x-values where the two curves intersect. Solving the equation x^2 = 2x gives us x = 0 and x = 2.

Using either method, we will integrate the volumes over the range [0, 2] and sum up the results:

If we choose the method of disks:
V = ∫[0,2] π * (2 - x)^2 * dx

If we choose the method of washers:
V = ∫[0,2] π * ((2 - x)^2 - (x^2)^2) * dx

By evaluating this integral, we can find the volume V of the solid obtained by rotating the region about the line x = 2.

To find the volume of the solid obtained by rotating the region enclosed by the curves y = x^2 and y = 2x about the line x = 2 using the method of disks or washers, we can follow these steps:

Step 1: Find the intersection points of the curves y = x^2 and y = 2x.
Set the two equations equal to each other:
x^2 = 2x
x^2 - 2x = 0
Factor out an x:
x(x - 2) = 0
So, we have two solutions: x = 0 and x = 2.

Step 2: Determine the bounds of integration.
Since the rotation is happening about the line x = 2, we need to find the interval of x-values that represents the region enclosed by the curves before rotation.
The curve y = x^2 is below y = 2x, so we only need to consider the portion of the curves where they intersect.
The intersection points are x = 0 and x = 2.
Thus, the bounds of integration are from x = 0 to x = 2.

Step 3: Determine the radius of each disk or washer.
For the method of disks, we consider a vertical slice at a given x-coordinate, and the radius of the disk is the distance from the x-axis to the curve y = 2x.
Therefore, the radius of each disk is given by r = 2x.

Step 4: Develop the integral.
The volume of each disk or washer is given by dV = π * r^2 * dx.
Substituting the radius r = 2x, the integral representing the volume becomes:
V = ∫[0,2] π * (2x)^2 dx.
Simplifying, we get:
V = ∫[0,2] 4πx^2 dx.

Step 5: Evaluate the integral.
Integrating the function 4πx^2 with respect to x, we get:
V = 4π * ∫[0,2] x^2 dx.
Evaluating the integral, we have:
V = 4π * [x^3/3] |[0,2].
Plugging in the upper and lower limits, we get:
V = 4π * (2^3/3 - 0^3/3).
Simplifying, we have:
V = 4π * (8/3).
This can be further simplified to:
V = (32π/3) cubic units.

Therefore, the volume of the solid obtained by rotating the region enclosed by y = x^2 and y = 2x about the line x = 2 using the method of disks or washers is (32π/3) cubic units.