Answer:
Explanation:
To find the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis, we can use the method of cylindrical shells.
The formula for the volume of a solid obtained by rotating a region bounded by the graph of a function f(x), the x-axis, x = a, and x = b about the y-axis is given by:
V = 2π ∫[a, b] x * f(x) dx
In this case, the function f(x) = (x - 4)^3, and the bounds of integration are a = 0 and b = 5.
Substituting these values into the formula, we have:
V = 2π ∫[0, 5] x * (x - 4)^3 dx
To evaluate this integral, we can expand the cubic term and then integrate:
V = 2π ∫[0, 5] x * (x^3 - 12x^2 + 48x - 64) dx
V = 2π ∫[0, 5] (x^4 - 12x^3 + 48x^2 - 64x) dx
Integrating each term separately:
V = 2π [1/5 x^5 - 3x^4 + 16x^3 - 32x^2] evaluated from 0 to 5
Now we can substitute the bounds of integration:
V = 2π [(1/5 * 5^5 - 3 * 5^4 + 16 * 5^3 - 32 * 5^2) - (1/5 * 0^5 - 3 * 0^4 + 16 * 0^3 - 32 * 0^2)]
Simplifying:
V = 2π [(1/5 * 3125) - 0]
V = 2π * (625/5)
V = 2π * 125
V = 250π
Therefore, the volume of the solid obtained by rotating the region bounded by the graphs y = (x - 4)^3, the x-axis, x = 0, and x = 5 about the y-axis is 250π cubic units.