use washer method
y=-4x+8,y=4x,x=0
To use the washer method, we need to have a region bounded by two curves and a vertical line. In this case, the region is formed by the lines y = -4x + 8 and y = 4x, and the vertical line x = 0.
Now, to find the volume of the solid formed when this region is revolved around the x-axis using the washer method, we can follow these steps:
1. Determine the limits of integration:
To do this, we need to find the x-values at which the two curves intersect. Setting the two equations equal to each other, we have:
-4x + 8 = 4x
8 = 8x
x = 1
So the limits of integration will be from x = 0 to x = 1, since the two curves intersect at x = 1.
2. Set up the integral for the volume:
The formula to find the volume using the washer method is:
V = π ∫[a, b] (R^2 - r^2) dx
Where R is the outer radius and r is the inner radius of the washers.
In this case, the outer radius will be the y-value of the curve y = -4x + 8, and the inner radius will be the y-value of the curve y = 4x.
So, the integral will be:
V = π ∫[0, 1] [(-4x + 8)^2 - (4x)^2] dx
3. Evaluate the integral:
Simplifying the integral, we have:
V = π ∫[0, 1] [(16x^2 - 64x + 64) - (16x^2)] dx
= π ∫[0, 1] (-64x + 64) dx
Integrating, we get:
V = π [-32x^2 + 64x] |[0, 1]
= π [(-32(1)^2 + 64(1)) - (-32(0)^2 + 64(0))]
= π [32]
= 32π
Therefore, the volume of the solid formed when the region bounded by the curves y = -4x + 8, y = 4x, and the line x = 0 is revolved around the x-axis using the washer method is 32π cubic units.