Final answer:
To find the volume of the solid obtained by rotating the region enclosed by the graphs about the y-axis, we can use the cylindrical shell method. Set up the integral using the intersection points and the larger and smaller functions. Simplify and evaluate the integral to find the volume.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region enclosed by the graphs about the y-axis, we need to set up the integral. The region is bounded by the graphs y = x³ and y = x¹/³. The volume can be calculated using the cylindrical shell method.
The formula for finding the volume using cylindrical shells is V = ∫[a,b] 2πx · (f(x) - g(x)) dx, where a and b are the x-values of the intersection points of the two curves, f(x) is the larger curve, and g(x) is the smaller curve.
First, we need to find the intersection points. Setting the two equations equal, we get x³ = x¹/³. Solving for x, we find x = 1. Now, we can set up the integral as V = ∫[0,1] 2πx · (x³ - x¹/³) dx. Simplifying and evaluating the integral will give us the volume of the solid.