Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.
y=6x^5, y=6x, x>or equal to 0
Find the volume V of this solid.
To find the volume V of the solid obtained by rotating the region bounded by the curves y = 6x^5 and y = 6x about the x-axis, you can use the method of cylindrical shells.
The equation of the curve y = 6x^5 intersects with the curve y = 6x at x = 1. Therefore, the bounds for integration would be from x = 0 to x = 1.
The first step is to express the differential element dx in terms of the variable y to set up the integral. We solve y = 6x^5 for x:
x = (y/6)^(1/5)
Next, we express the equation of the curve y = 6x as x = y/6.
The radius of each cylindrical shell at a given height y is the distance between these two curves:
r = (y/6)^(1/5) - (y/6).
The height of each cylindrical shell is given by the differential element dx: h = dx.
The volume of each cylindrical shell is given by:
dV = 2πrh dx = 2π((y/6)^(1/5) - (y/6)) dx.
To find the total volume, we integrate this expression with respect to x over the interval [0, 1]:
V = ∫[0,1] dV
= ∫[0,1] 2π((y/6)^(1/5) - (y/6)) dx.
Evaluating this integral will give you the volume V of the solid.