Final answer:
To find the volume of the solid formed by rotating the given curves about the y-axis, express x as a function of y, determine the limits of integration from intersecting points of the curves, and use the integral formula for volume by revolution.
Step-by-step explanation:
To find the volume V of a solid of revolution obtained by rotating the region bounded by the curves y² = 2x and x = 2y about the y-axis, we can use the disk or washer method. To set up the integral, first express x as a function of y since we are revolving around the y-axis. From y² = 2x, we get x = y² / 2. From x = 2y, we get y = x / 2 or x = 2y.
Next, we need to determine the limits of integration which are the points where the curves intersect. Setting y² / 2 equal to 2y, we get y² = 4y, or y(y - 4) = 0, yielding y = 0 and y = 4 as the limits of integration.
Finally, the volume by revolution is found using the integral formula V = π ∫_{a}^{b} (outer radius)^2 - (inner radius)^2 dy. In this case, there is no inner radius because we do not have a hole in the solid, so the volume is V = π ∫_{0}^{4} (y² / 2)^2 dy. Upon integrating, we find the volume V of the solid.