Final answer:
The volume V of the solid, created by rotating the region bounded by the curves y=6-x and y=0 about the x-axis from x=0 to x=1, can be calculated using the disk method, resulting in V approximately equal to 30.33π cubic units.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves y = 6 - x and y = 0, between x=0, and x = 1 about the x-axis, we use the disk method. The area of a typical disk at a given value of x is A(x) = π (6-x)^2. The volume is then the integral of A(x) with respect to x from 0 to 1.
V = ∫_0^1 π (6 - x)^2 dx
Let's compute the integral:
V = π ∫_0^1 (36 - 12x + x^2) dx
V = π [36x - 6x^2 + (1/3)x^3] from 0 to 1
V = π [36(1) - 6(1) + (1/3)(1) - (36(0) - 6(0) + (1/3)(0))]
V = π (36 - 6 + 1/3)
The calculated volume V will then be approximately: V = 30.33π cubic units.