Find the volume of the solid formed by rotating the region enclosed by
x=0, x=1, y=0,y= 6 +x^{8}
about the y-axis.
To find the volume of the solid formed by rotating the region bounded by the curves x=0, x=1, y=0, and y=6+x^8 about the y-axis, we can use the method of cylindrical shells.
1. First, let's visualize the region bounded by the given curves. The graph of y=6+x^8 looks like a curve that starts at y=6 when x=0 and rises rapidly as x increases. The region bounded by the curve and the lines x=0, x=1, and y=0 will therefore be a shape similar to a triangle.
2. Next, we need to determine the height and radius of each cylindrical shell. Since we are rotating the region about the y-axis, the height of each cylindrical shell will be the change in y from the lower curve to the upper curve at a specific x-coordinate. In this case, the upper curve is y=6+x^8 and the lower curve is y=0. So, the height of each cylindrical shell is (6+x^8) - 0, which simplifies to just (6+x^8).
The radius of each cylindrical shell will be the x-coordinate value at a specific point on the curve. As we rotate the region about the y-axis, the x-coordinate is essentially the distance from the y-axis to the curve. Therefore, the radius is the x-coordinate value itself, which is just x.
3. Now, we can express the volume of each cylindrical shell. The volume of a cylindrical shell is given by the formula V = 2πrh, where r is the radius and h is the height. In this case, the volume of each cylindrical shell is equal to 2πx(6+x^8).
4. To find the total volume, we need to integrate the volume of each cylindrical shell over the interval of x=0 to x=1. The integral form of the volume equation becomes V = ∫[0,1] 2πx(6+x^8) dx.
5. Integrate the equation to find the volume:
V = ∫[0,1] 2πx(6+x^8) dx
= 2π ∫[0,1] (6x + x^9) dx
= 2π [(3x^2/2) + (x^10/10)] evaluated from 0 to 1
= 2π [ (3/2) + (1/10) - 0 ]
= 2π [ (3/2) + (1/10) ]
= 2π [ (15/10) + (1/10) ]
= 2π [ 16/10 ]
= 32π/10
Therefore, the volume of the solid formed by rotating the region bounded by x=0, x=1, y=0, and y=6+x^8 about the y-axis is 32π/10 (or 16π/5) cubic units.