Question

Evaluate the line integral 3x ds, where C is the quarter-circle x^2+y^2=4 from (2, 0) to (0, 2).

Answer 1

Parameterize \(C\) by \(x(t)=2\cos t,\;y(t)=2\sin t, \text{ with }0\le t\le\frac{\pi}{2}.\)
I'm not sure what the "ds" means, though. Is that supposed to mean \(\sqrt{1+\left(\frac{dy}{dt}\right)^2}\), or something like that?

Answer 2

You're right.

Answer 3

Are you absolutely sure? Because it could also be \(\frac{dx}{dt}, \text{ or }\frac{dy}{dx}.\)

Answer 4

^^ instead of \(\frac{dy}{dt},\) I mean.

Answer 5

But where did you get the x(t)=2/sin(t) and y(t)=2/cos(t)?

Answer 6

Those are the parametric equations of a circle with radius 2 (centered about the origin). They describe the path C.

Answer 7

What's the next step I should do?

Answer 8

Set up your integral. Depending on what dS is supposed to be, I'm pretty sure it'd be
\[\large\int_C3x\;dS=\int_0^{\frac{\pi}{2}}3\cos t\sqrt{\cdots}\;dt\]

Answer 9

But how to find the ds?

Answer 10

Ah, according to this link ( http://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx),
\[dS=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\;dt\]
In my earlier posts, I was stuck in an arc-length-mindset.

So, the integral is
\[\large\int_0^{\frac{\pi}{2}}3\cos t\sqrt{\left(-2\sin t\right)^2+\left(2\cos t\right)^2}\;dt\]

Answer 11

Thanks.