Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.
y = 6 x^6 , y = 6 x , x >= 0
Find the volume V of this solid.
The curves intersect at x = 0 and x = 1. The region bounded between those curves has y-separation of 6(x-x^6).
For the total enclosed area, integrate that function times dx from x=0 to x=1.
I forgot that you wanted the volume of the solid obtained by rotating the curves about the x axis. This changes the formula to
Integral of 36 pi (x^2 - x^12) dx
...0 to 1
To find the volume V of the solid, obtained by rotating the region bounded by the curves y = 6 x^6 and y = 6 x about the x-axis, we can use the method of cylindrical shells.
The volume V can be obtained by integrating the area of the cylindrical shell over the range of x that corresponds to the region bounded by the curves.
First, let's find the points of intersection of the curves y = 6 x^6 and y = 6 x:
6 x^6 = 6 x
Dividing both sides by 6:
x^6 = x
Since the inequality x >= 0 is given, we can multiply both sides by x:
x^7 = x^2
Now we have two cases to consider:
Case 1: x = 0
In this case, the region is bounded by the y-axis and the curve y = 6 x, so the volume contribution at x = 0 is 0.
Case 2: x ≠ 0
In this case, we can divide both sides of the equation x^7 = x^2 by x^2 (because x ≠ 0):
x^5 = 1
Taking the fifth root of both sides:
x = 1
Now we have the limits of integration for calculating the volume of the solid using cylindrical shells. The integral that represents the volume V is:
V = ∫[0,1] 2πx (6x - 6x^6) dx
Simplifying the integrand:
V = ∫[0,1] 12πx (1 - x^5) dx
Now we can integrate:
V = 12π ∫[0,1] (x - x^6) dx
V = 12π [(x^2/2) - (x^7/7)] [0,1]
V = 12π [(1/2) - (1/7)] = 12π [7/14 - 2/14]
V = 12π (5/14) = 60π/7
Therefore, the volume V of the solid obtained by rotating the region bounded by the curves y = 6 x^6 and y = 6 x about the x-axis is 60π/7.