## Integral around half an ellipse

Calculate the integral $\displaystyle \int_{\gamma} (x_1 x_2 dx_1 + x_1^2 dx_2)$ in each of the following two cases:

1. $\displaystyle \gamma$ is the bottom half of the ellipse $\displaystyle 2x_1^2 + 3x_2^2 = 8$, traversed from the point (2,0) to the point (-2,0);

2. $\displaystyle \gamma$ is the complete ellipse $\displaystyle 2x_1^2 + 3x_2^2 = 8$, traversed in the counterclockwise direction.

Also, write the integral as the integral $\displaystyle \int_{\gamma} \mathbf{f}(\mathbf{x}) \cdot \mathbf{\tau} ds$ with respect to arc length of the dot product of a vector field $\displaystyle \mathbf{f}(\mathbf{x})$ with the unit tangent vector $\displaystyle \mathbf{\tau}$ to the curve $\displaystyle \gamma$.

I got 0 for both my integrals. The way I did them was parametrize the curve in two different ways (first one using t, second one using $\displaystyle \theta$), but they both gave me 0. I am not sure if this should be correct, can someone please confirm? The vector field for the second part of the problem is obviously $\displaystyle \mathbf{f}(\mathbf{x}) = (x_1x_2, x_1^2)$, right? Is it a trick question?