Final answer:
To find the volume of the solid obtained by rotating the region bounded by the curves y=x² and y=4x about the line y=x², you can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves y=x² and y=4x about the line y=x², we can use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula V = 2πrhΔx, where r is the distance from the axis of rotation to the shell, h is the height of the shell, and Δx is the thickness of the shell.
We can express the equations of the curves in terms of x as y = x² and y = 4x. Solving for x, we get x = y^(1/2) and x = y/4. The distance from the axis of rotation (y=x²) to the shell is r = x - x², and the height of the shell is h = y/4 - y^(1/2).
Now, we can integrate the volume over the range where the curves intersect, which is from x = 0 to x = 4. The integral expression for the volume is as follows:
V = ∫0⁴ 2π(x-x²)(y/4 - y^(1/2)) dx
Simplifying and evaluating this integral will give us the volume of the solid.