Final answer:
To sketch the region bounded by the curves x=2y³, x=2, and y=0, plot the curves on a coordinate plane. To find the volume of the solid generated by revolving this region about the y-axis, use the method of cylindrical shells and integrate.
Step-by-step explanation:
To sketch the region bounded by the curves x = 2y³, x = 2, and y = 0, we can first plot these curves on a coordinate plane. The first curve, x = 2y³, is a curve that represents a relationship between x and y. As we substitute different values for y, we can find the corresponding x values. The second curve, x = 2, is a vertical line that intersects the y-axis at x = 2. The third curve, y = 0, is the x-axis itself.
Once we have sketched the region, we can find the volume of the solid generated by revolving this region about the y-axis. To do this, we can use the method of cylindrical shells. We can break the region into infinite thin vertical strips, each with a width of Δy. Each strip represents a cylindrical shell with radius x and height Δy. The volume of each shell is given by V = 2πxyΔy. Then, we can integrate from y = 0 to the highest y-value of the curve x = 2y³. The integral expression for the volume is given by V = ∫0^(highest y-value) 2πxy dy.