Final answer:
To find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the x-axis, we use the method of cylindrical shells. By setting the two curves equal to each other, we find the points of intersection. Then, we set up a definite integral to find the volume, substituting the y-values for each curve and integrating. The resulting volume is 0.
Step-by-step explanation:
To find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the x-axis, we need to use the method of cylindrical shells. First, we need to find the points of intersection of the two curves by setting them equal to each other:
x² = 4x - x²
Simplifying the equation, we get:
2x² - 4x = 0
Factoring out a common factor of 2x, we get:
2x(x - 2) = 0
Setting each factor equal to zero, we find that x = 0 or x = 2.
Now, we can set up the definite integral to find the volume of the solid:
V = ∫[(2π)(radius)(height)] dx
The radius is the distance between the x-axis and the curve, which is given by r = y. The height is the differential length element dx. We integrate from x = 0 to x = 2:
V = ∫[(2π)(y)(dx)] from 0 to 2
Now, we substitute the y-values for each curve:
V = ∫[(2π)(x²)(dx)] from 0 to 2 - ∫[(2π)(4x - x²)(dx)] from 0 to 2
Simplifying and integrating, we get:
V = 2π[(x³/3)] from 0 to 2 - 2π[(2x² - x³/3)] from 0 to 2
V = (8π/3) - (8π/3) = 0
Therefore, the volume of the solids generated by revolving the regions bounded by the graphs of the equations about the x-axis is 0.