(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

What is [tex]\int_{\gamma} xy dx + x^2 dy[/tex] in each of the following cases?

1. [tex]\gamma[/tex] is the lower half of the curve [tex]2x^2 + 3y^2 = 8[/tex], traveled from [tex](2,0)[/tex] to [tex](-2,0)[/tex].

2. [tex]\gamma[/tex] is the full curve [tex]2x^2 + 3y^2 = 8[/tex], traveled counterclockwise.

2. Relevant equations

The line integral formula, I suppose. The fact that the integral can be expressed as the dot product of the vector field [tex](xy, x^2)[/tex] with the unit tangent vector to the curve can also be helpful.

3. The attempt at a solution

I parametrized the curves for (1) and (2) in different ways, viz.

1. [tex]x = t, y = -2\sqrt{\frac{2}{3}\left(1 - \frac{t^2}{4}\right)}[/tex].

2. [tex]x = 2\cos{\theta}, y = 2\sqrt{2/3}\sin{\theta}[/tex].

Then standard integration rules, but I came up with 0 for both the answers. Am I correct?

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# Homework Help: Line integral around an ellipse

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