Final answer:
To sketch the region bounded by the curves x=y2 and 2x=y and find the volume of the solid generated by revolving this region about the y-axis using the shell method, we first find the points of intersection between the two curves and sketch the region. Then, we divide the region into vertical cylindrical shells and calculate the volume of each shell. Finally, we integrate the expression from y=0 to y=2 to find the total volume of the solid.
Step-by-step explanation:
To sketch the region bounded by the curves x=y^2 and 2x=y, we can start by finding the points of intersection between the two curves. Setting the equations equal to each other, we get y^2 = 2y, which simplifies to y(y - 2) = 0. This gives us two critical points: y = 0 and y = 2. Next, we can plot the curves and the points of intersection on a coordinate plane to sketch the region.
Now, to find the volume of the solid generated by revolving this region about the y-axis using the shell method, we divide the region into an infinite number of vertical cylindrical shells. The height of each shell is given by the difference in the y-coordinates of the upper and lower curves, which is 2 - y^2. The radius of each shell is the x-coordinate, which is y^2. The volume of each shell is then given by V = 2πy^3(2 - y^2)dy. We integrate this expression from y = 0 to y = 2 to find the total volume of the solid.