Hello, nmq3b!
Can you sketch the region, or do you have a graphing calculator?
A good sketch is absolutely essential.
1. Use the method of cylindrical shells to find the volume of the solid obtained by rotating
about the yaxis, the region R bounded by the curve $\displaystyle y\:=\:2x^2x^3$ and the xaxis. Code:

*  *
 *:::::*
*  *:::::::*
*  *::::::::::
   *       *  
 2
 *
 *
Shells Formula: .$\displaystyle V \;=\;2\pi\int^b_a\text{(radius)(height)}\,dx$
In this problem: .$\displaystyle \text{radius} \:=\:x,\;\text{height} \:=\:2x^2x^3$
Hence: .$\displaystyle V \;=\;2\pi\int^2_0x(2x^2x^3)\,dx$
I assume you can finish it . . .
2. Use the method of cylindrical shells to find the volume of the solid obtained rotating
the region R boounded by the curves $\displaystyle y\:=\:x^3,\;y = 0,\;x = 2$ about the line $\displaystyle x = 3$ Code:
 * :
  :
 * :
 *: :
 .*::: :
    *    +  +  
*  2 3
* 
* 
This time we have: .$\displaystyle \text{radius } = 3x,\;\text{height} = x^3$
Hence: .$\displaystyle V \;=\;2\pi\int^2_0(3x)x^3\,dx$