Final answer:
To find the volume of the solid obtained by rotating the region about the y-axis, we can use the disk method. The disk method involves dividing the region into small discs, finding the volume of each disc, and then summing up the volumes to get the total volume.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region about the y-axis, we can use the disk method. The disk method involves dividing the region into small discs, finding the volume of each disc, and then summing up the volumes to get the total volume.
First, we need to find the limits of integration. The region is bounded by y = 2x² - x³ and y = 0. Setting the equations equal to each other, we find that 2x² - x³ = 0, which gives us x = 0 and x = 2 as the limits of integration.
Next, we need to express the equation in terms of x instead of y. Rearranging the equation, we have y = x³ - 2x². Now we are ready to calculate the volume. The volume of each disc is given by V = πr²h, where r is the distance from the disc to the axis of rotation (in this case, the y-axis), and h is the height of the disc. Since we are rotating about the y-axis, the height of each disc is given by h = y. The distance from the disc to the axis of rotation is the x-coordinate of the disc, which is x. So, the volume of each disc is V = πx²(y). Integrate this expression from x = 0 to x = 2 to find the total volume.