Final Answer:
1. (a) The volume of the solid formed by rotating the region (R) about the x-axis using (x) as the variable of integration is
.
(b) The volume of the solid formed by rotating the region (R) about the x-axis using (y) as the variable of integration is

2. (a) The volume of the solid formed by rotating the region (R) about the line (x = 3) using (x) as the variable of integration is
.
(b) The volume of the solid formed by rotating the region (R) about the line (x = 3) using (y) as the variable of integration is

Step-by-step explanation:
1. (a) To find the volume using (x) as the variable of integration, integrate the difference of the outer and inner functions squared,
. In this case, it's
![\(\pi \int_(0)^(2) \left[ (3x - 2)^2 - x^4 \right] \,dx\).](https://img.qammunity.org/2024/formulas/mathematics/high-school/khqbahsjby8ejpb0ygi6855sgy4rwqle3l.png)
(b) Using (y) as the variable of integration, the formula becomes
, where (F) and (G) are the inverse functions of (f) and (g). Here, it's
![\(\pi \int_(-2)^(4) \left[ √(y + 2) - (y)/(3) \right]^2 \,dy\).](https://img.qammunity.org/2024/formulas/mathematics/high-school/ws6mb71eqlk4qxas8ee0i48cv6zrjfirgp.png)
2. (a) When rotating about (x = 3), modify the integrand to
. For this scenario, it's
![\(\pi \int_(0)^(2) \left[ (3x - 2)^2 - (x - 1)^2 \right] \,dx\).](https://img.qammunity.org/2024/formulas/mathematics/high-school/pwyfwsriivebzrry9up4b0zm8k432fgegf.png)
(b) Using (y) as the variable, the formula becomes
, where (F) and (G) are the inverse functions of (f) and (g). Here, it's
![\(\pi \int_(-2)^(4) \left[ √(y + 2) - (y)/(3) - 3 \right]^2 \,dy\).](https://img.qammunity.org/2024/formulas/mathematics/high-school/gj232tkh8uel4nvf03ptecpm161y6z9q12.png)