Final answer:
To find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis, we can use the method of cylindrical shells. The volume of the solid is found by integrating the area of each cylindrical shell along the x-axis. The volume of the solid is 8π.
Step-by-step explanation:
To find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis, we can use the method of cylindrical shells. First, let's find the intersection points of the lines and curves:
y = -4x + 8 and y = 4x
Setting these two equations equal to each other, we get:
-4x + 8 = 4x
Simplifying, we have:
8 = 8x
x = 1
So, the intersection point is (1,4).
To find the volume, we need to integrate the area of each cylindrical shell along the x-axis. The height of each shell is given by the difference between the curves y = -4x + 8 and y = 4x, and the radius of each shell is given by the distance between the x-axis and the point on the curve y = -4x + 8.
The integral to find the volume is:
V = ∫[0,1] 2πx(-4x + 8 - 4x) dx
Simplifying the integral, we get:
V = ∫[0,1] -8πx^2 + 16πx dx
Integrating, we have:
V = [-8π(x^3/3) + 16π(x^2/2)]10
V = -8π(1/3) + 16π(1/2)
V = -8π/3 + 8π
V = 24π/3
V = 8π
Therefore, the volume of the solid is 8π.