Surface area of (1/3)(y^2 + 2)^(3/2) ?

Question

Update : I got the wrong answer for this question. 

Question: Find the exact surface area obtained by rotating the curve (1/3)(y^2 + 2)^(3/2) about the x-axis 1 <= y <= 2. 

What I did: 
dx/dy = (1/2)(y^2+2)^0.5 

Area = ∫ 2pi*y*sqrt(1+ (1/4)(y^2 + 2) ) dy from 1 to 2 

= 2pi ∫ y*sqrt(4+y^2+2)/2 dy 
=pi ∫ y*sqrt(y^2+6) dy 
=pi [(2/3)(y/2y)(6+y^2)^(3/2)] from 1 to 2 
= (pi/3)(10*sqrt(10) + 7*sqrt(7)) 

But the textbook answer is (21pi/2). :( 

Where did I go wrong?

Answer 1

 Surface area of

If the curve described as x = g(y) , a  ≤ y ≤ b then formula for surface area becomes.

In this case a = 1 and b = 2

Surface area is 21pi/2 square units.