Find the volume generated. The region is bounded by y = 4x-x^2 and y = 8x-2x^2, and is rotated about x = 2. State which method (washer, disk, cylindrical shell) is used
To find the volume generated by rotating the region bounded by the curves y = 4x - x^2 and y = 8x - 2x^2 about the line x = 2, the cylindrical shell method can be used.
The first step is to find the points of intersection between the two curves.
Setting 4x - x^2 = 8x - 2x^2, we can rearrange to obtain:
0 = 4x - 8x + x^2 - 2x^2
0 = x^2 - 4x
Factoring out an x:
0 = x(x - 4)
Setting each part equal to zero, we find two critical points: x = 0 and x = 4.
Now, we can integrate to find the volume. Since we are using the cylindrical shell method, the integral will be with respect to y.
The height of each cylindrical shell is given by h = y - (4x - x^2), which simplifies to h = y + x^2 - 4x.
The radius of each cylindrical shell is given by r = x - 2, since we are rotating about the line x = 2.
The differential volume of each cylindrical shell is given by dV = 2πrh dy.
Integrating this expression from y = 4x - x^2 to y = 8x - 2x^2 and from x = 0 to x = 4, we have:
V = ∫(4,8)∫(0,4) 2π(rh) dy dx
V = ∫(4,8)∫(0,4) 2π(y + x^2 - 4x)(x - 2) dy dx
By evaluating this double integral, we can find the volume generated by the rotation.
To find the volume generated by rotating the region bounded by the two curves, y = 4x - x^2 and y = 8x - 2x^2, about the line x = 2, we can use the method of cylindrical shells.
Here's how you can calculate the volume using this method:
1. First, let's find the intersection points of the two curves. Set the two equations equal to each other:
4x - x^2 = 8x - 2x^2
2. Simplify the equation:
-x^2 + 4x = -2x^2 + 8x
3. Rearrange the equation to make it equal to zero:
x^2 -4x + 4x = 0
4. Factorize the equation:
x(x - 4) = 0
So, there are two intersection points: x = 0 and x = 4.
5. Next, find the equations of the curves when they are rotated around x = 2. We need to find the distance between the original x-axis and the line of rotation, which is x = 2. The distance is 2 units.
6. To form cylindrical shells, we consider an infinitesimally thin shell at a distance x from the line x = 2. The height of the shell will be the difference between the two curves at that x-value: (8x - 2x^2) - (4x - x^2).
7. The radius of the shell will be the distance from the line of rotation (x = 2) to the x-value of the shell, which is 2 - x.
8. Now, we can set up the integral to find the total volume:
V = ∫[a, b] (2π * (2 - x) * ((8x - 2x^2) - (4x - x^2))) dx
Where 'a' and 'b' are the x-values of the intersection points, which are 0 and 4, respectively.
9. Simplify the integral and evaluate it between the limits a and b to find the volume generated.
In summary, we can use the method of cylindrical shells to find the volume generated by rotating the given region about x = 2.