your method using cylindrical shells is correct.
you have to use disks from y = 0 to y = 4 and the washers from y = 4 to y = 8 ... i.e. , two integral expressions
I've been trying to calculate the volume of a solid of revolution. The 2D graph of it is the region bounded by y=x^3, y= 4+2x, and the y axis. This is rotated around the y axis. I tried both the washer and cylinder methods and each gives a different answer. Which one is the real volume, and where did I go wrong?
Here's a graph of the problem and my results for each method: