Parameterize the path by
r (x) = (x, x + 2)
with 0 ≤ x ≤ 2. Then
r' (x) = (1, 1) ==> ||r' (x)|| = √(1² + 1²) = √2
and the line integral is
[tex]\displaystyle\int_Lxy\,\mathrm ds=\sqrt2\int_0^2x(x+2)\,\mathrm dx=\boxed{\frac{20\sqrt2}3}[/tex]
