Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = x, y = 0, x = 2, x = 6; about x = 1
To find the volume V of the solid obtained by rotating the region bounded by the curves y = x, y = 0, x = 2, and x = 6 about the line x = 1, we can use the method of cylindrical shells.
1. First, let's plot the region and the axis of rotation to better understand the problem. The region bounded by the curves y = x, y = 0, x = 2, and x = 6 is a triangular region.
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1 2 6
2. Next, we need to determine the height of each cylindrical shell. Since we are rotating the region around x = 1, the distance between the axis of rotation and the curve y = x represents the radius of each shell. The height of each shell is the difference between the upper and lower boundaries of the region, which is 0 and x.
3. To set up the integral for the volume of a single shell, we consider a small segment of x within the region, denoted as dx. Thus, the volume of each cylindrical shell is given by dV = 2π(x - 1) dx.
4. To find the limits of integration, it is important to note that the region starts at x = 2 and ends at x = 6. Therefore, the integral for the volume becomes:
V = ∫[2, 6] 2π(x - 1) dx.
5. Integrating the expression, we have:
V = 2π ∫[2, 6] (x - 1) dx
= 2π [Δx^2/2 - Δx] evaluated from 2 to 6
= 2π [(6 - 1)^2/2 - (2 - 1)^2/2]
= 2π (25/2 - 1/2)
= 24π.
Thus, the volume V of the solid obtained by rotating the region bounded by the curves y = x, y = 0, x = 2, and x = 6 about the line x = 1 is 24π cubic units.