Final answer:
To find the volume of the solid formed by rotating the region enclosed by x=8y and y³=x around the y-axis, one can use the cylindrical shells method and integrate from y = 0 to y = 2. The volume is calculated through the integral of 2πy(8y-y³)dy from 0 to 2.
Step-by-step explanation:
The question seeks to calculate the volume of a solid of revolution, which is created by rotating a region enclosed by two equations - x = 8y and y³ = x - around the y-axis. To find this volume, we can use the method of cylindrical shells. The volume of a thin shell with radius y, height x, and thickness dy is given by V = 2πyhdy. The height of the shell (h) in our case is the difference between the two functions: x from x = 8y and x from y³ = x.
Integrating from y = 0 to the point where the curves intersect will yield the total volume. The intersection points are determined by setting x=8y equal to y³ = x, which yields y = 2. Thus, we integrate the volume of the cylindrical shells from y = 0 to y = 2. The integral is ∠ 2πy(8y - y³)dy from 0 to 2.
Solving the integral, we get the formula for volume: V = 2πࢬ (8y² - y⁴)dy from 0 to 2, which, after evaluating, gives us the final volume of the solid of revolution.