I believe 2pi/15 is correct

Draw a diagram and you'll see you need 2 integrals

between x = 0 and 1/2 you have disks

V= (pi)integral(4x^4dx) = pi/40

between x =1/2 and 1 you have washers

V = pi integral (4x^4 - (4x-2)^2)dx) =13pi/120

Adding the 2 you get 16pi/120 = 2pi/15