$$\int_{γ,p} (2x \ y \ dx + (x^3 + 3z) \ dy + 3y \ dz)$$ where $$γ = [(0, 0, 0),(0, 1, 3)] ∪ \{ (x, y, z) ∈ \mathbb{R^3}|y = 0, \ x^2 + (z − 3)^2 = 9, \ x ≤ 0\} ∪ [(0, 0, 6),(1, 1, 6)]$$ and p is the orientation so that $(0,1,3)$ is the first end.
It looks to me as if each term is meant to be integrated over the range of only one variable: am I wrong? E.g., $$\int_{γ,p} 2x \ y \ dx + \int_{γ,p} (x^3 + 3z) \ dy + \int_{γ,p} 3y \ dz $$ and the first integral becoming, for instance, $$\int_{γ_!,p} 2x \ y \ dx + \int_{γ_2,p} 2x \ y \ dx + \int_{γ_3,p} 2x \ y \ dx = \int_0^0 0 \ ydx + \int_0^3 2x \ 0 \ dx + \int_0^1 2x \ x \ dx = x^2|_0^1 = 1 $$
Am I off? If so, why?
The definition of the scalar product and the linearity of the line integral give for some vector function $A$: $$ \int\limits_C A \cdot du = \int\limits_C \left(\sum_i A_i \, e_i \right) \cdot du = \sum_i \int\limits_C A_i \, e_i \cdot du = \sum_i \int\limits_C A_i \, dx_i $$ where the $e_i$ are the canonical base vectors.