Answer:
the volume of the solid generated when R is revolved about the y-axis is π/18.
Explanation:
To find the volume of the solid generated when the region R is revolved about the y-axis, we can use the method of cylindrical shells.
The region R is bounded by the curves y = x^7 - x^10 and y = 0. To find the limits of integration, we need to find the x-values where the two curves intersect.
Setting the two equations equal to each other, we have:
x^7 - x^10 = 0
Factoring out x^7, we get:
x^7(1 - x^3) = 0
This equation gives us two solutions: x = 0 and x = 1.
Therefore, the limits of integration for x are 0 and 1.
Now, let's set up the integral for the volume using the cylindrical shell method:
V = ∫[a,b] 2πx * h(x) * dx
where a = 0, b = 1, h(x) is the height of the shell at x, and dx represents the differential width of the shell.
The height of each cylindrical shell, h(x), is given by the difference between the two curves:
h(x) = (x^7 - x^10) - 0 = x^7 - x^10
Substituting this into the integral, we have:
V = ∫[0,1] 2πx * (x^7 - x^10) * dx
To evaluate this integral, we can expand the expression inside the integral and then integrate term by term.
V = 2π ∫[0,1] (x^8 - x^11) * dx
= 2π [ (1/9)x^9 - (1/12)x^12 ] |[0,1]
= 2π * [(1/9) - (1/12)]
= 2π * [(4/36) - (3/36)]
= 2π * (1/36)
= π/18
Therefore, the volume of the solid generated when R is revolved about the y-axis is π/18.