Final answer:
To calculate the volume of the rotated solid, we use the washer method to integrate the difference between the squares of the outer and inner radii of the washers formed about the y-axis from the boundaries set by the intersecting curves.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by y²=x and x=2y around the y-axis, we can use the washer method. The washer method takes into account the area of the outer radius squared minus the inner radius squared multiplied by π and integrated over the bounds of y.
First, we need to express both equations in terms of y. We already have x=2y, and for y²=x, solving for x gives us x=y². The bounds of integration are where these two curves intersect, which occurs at y = 0 and y = 2. Next, we set up the integral:
V = π∫_from 0_to 2 (outer radius) ² - (inner radius) ² dy
The outer radius is from the curve x=2y which is simply 2y, and the inner radius is from the curve y², so our integral becomes:
V = π∫_from 0_to 2 (2y)² - (y²)² dy
After computing the integral, we will obtain the volume of the solid. As we go through the integration process, remember to raise each function to the second power and integrate with respect to y.