Surface integral or Divergence Theorem confused?

[abrowaqas]

Homework Statement

Find the Volume
∫∫ xy DA
where R is the region bounded by by the line y=x-1 and the parabola y^2=2x+6.

Homework Equations

∫∫ xy dx dy

The Attempt at a Solution

first i found the intersection of the above equations . which is (5,4) to (-1,-2) . then i simple put the values in the limits of the integral
i-e

∫(from y= -2 to y=4) ∫(from x= y+1 to x= (y^2-6)/2 ) xy dx dy

and solve it and finaly got 52/3.

is this the right method and limits are correct or not ?
or i have to use here divergence theorem

can somebody explain the word VOLUME in the question ?

 

[HallsofIvy]

The word "volume" is an error. Clearly that is a two dimensional figure so it has area not volume.

 

[abrowaqas]

so that means my method is correct?

but someone says

If the problem really is to find a "volume", it should read ∫ ∫ |xy| dA because the integrand is not positive everywhere on the region given.

Can't be sure, but that's probably not what the author intended. He just wants you to integrate xy over the region. It has the form ∫ ∫ z(x, y) dA which is a volume integral if z(x, y) is non-negative over the region.