Very quick question about parametrizing an ellipse (for line integral).

Question

Here's the integral I'm solving: $\oint_C(7x+8y)dy$ (Clockwise). $C: 4x^2+9y^2=1$
I'm asked first to find a parametrization for the ellipse that makes sure $t\in[0,2\pi]$.
So what I picked was $x=\frac{1}{2}\cos(t), y=\frac{1}{3}\sin(t)$.
But I got the wrong answer, and the only difference was that the parametrization they picked is $x=-\frac{1}{2}\cos(t), y=\frac{1}{3}\sin(t)$.

And now I'm super confused of why there's a minus and why it matters? I tried to think that it comes from making sure $t\in[0,2\pi]$, but my parametrization also does that, and I can't understand why there's a minus and why it's important.

Would appreciate any feedback, thanks in advance!

Answer 1

The question says clockwise. Coming to parametrization,

$x=-\frac{1}{2}\cos(t), y=\frac{1}{3}\sin(t), t \in [0, 2\pi]$

As $t$ goes from $0$ to $2\pi$, you in effect move from $\pi$ to $-\pi$ on the curve in clockwise direction.

Rewrite it as $x=\frac{1}{2}\cos(\pi - t), y=\frac{1}{3}\sin(\pi-t), t: 0 \to 2\pi \ , \ $ to understand better.

You do not necessarily need to choose the parametrization they chose.

$x=\frac{1}{2}\cos(- t), y=\frac{1}{3}\sin(-t) \ $ i.e $x=\frac{1}{2}\cos(t), y=-\frac{1}{3}\sin(t), t: 0 \to 2\pi \ $ works too for an example.