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

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

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

Also, write the integral as the integral \int_{\gamma} \mathbf{f}(\mathbf{x}) \cdot \mathbf{\tau} ds with respect to arc length of the dot product of a vector field \mathbf{f}(\mathbf{x}) with the unit tangent vector \mathbf{\tau} to the curve \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 \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 \mathbf{f}(\mathbf{x}) = (x_1x_2, x_1^2), right? Is it a trick question?