Find the volume of the solid obtained by rotating the region bounded by the curves y=x^6, y=1 about the line y=4
To find the volume of the solid obtained by rotating the region bounded by the curves y = x^6, y = 1 about the line y = 4, we can use the method of cylindrical shells.
The first step is to sketch the region and identify the bounds of integration. In this case, the region is bounded by two curves: y = x^6 and y = 1. The curves intersect at the points where x^6 = 1, so we need to solve the equation x^6 = 1 to find the bounds.
Taking the sixth root of both sides, we get x = ±1. Since the region lies between x = -1 and x = 1, these will be our bounds of integration.
Next, we need to set up the integral that represents the volume of the solid. The volume of a cylindrical shell is given by the formula:
V = ∫(2πrh)dx
where r is the radius of the shell and h is the height (or the differential length) of the shell. In this case, the radius is the distance between the line y = 4 and the curve y = x^6, which is 4 - x^6.
The height of the shell is dx, as we are integrating with respect to x. Hence, the integral representing the volume is:
V = ∫(2π(4 - x^6)) dx,
with the bounds of integration being x = -1 to x = 1.
Evaluating this integral will give you the volume of the solid obtained by rotating the region bounded by the curves y = x^6, y = 1 about the line y = 4.