# Paramtrization and Line Integral

• Jul 1st 2010, 03:36 PM
keysar7
Paramtrization and Line Integral
a) Parametrize the curve $x^{\frac{2}{3}}+y^{\tfrac{2}{3}}=a^{\tfrac{2}{3}}\ \$ in the standard counterclockwise sense.

b) Evaluate $\text{F(x) = (}\tfrac{-1}{\sqrt[{3}]{{y}}}\text{ ,}\tfrac{1}{\sqrt[{3}]{{x}}}\text{ )}$ over one complete transversal of the above curve.

_____________________

a)

$x=a\text{ }\cos ^{3}t$
$y=a\text{ }\sin ^{3}t$
$0\text{ }\leq t\leq \text{ }2\pi$

b) Since the field is not conservative, we have to find the line integral (i.e. the answer is not zero)... My final answer is $-6\pi a^{\tfrac{2}{3}}$

However, I think the negative sign is not correct because the force field is along the path of motion, meaning positive would make more sense...

What do you think?
• Jul 3rd 2010, 05:04 PM
keysar7
I'm starting to doubt my answer. And nobody is offering any help...

So here's another question.

1. Is there a way I could verify my parametrization? Is there a good online grapher that does both regular and parametric graphing?

And

2. Even though I did the line integral by hand, is there an online integrator that could help in this case?

• Jul 3rd 2010, 05:54 PM
mr fantastic
Quote:

Originally Posted by keysar7
a) Parametrize the curve $x^{\frac{2}{3}}+y^{\tfrac{2}{3}}=a^{\tfrac{2}{3}}\ \$ in the standard counterclockwise sense.

b) Evaluate $\text{F(x) = (}\tfrac{-1}{\sqrt[{3}]{{y}}}\text{ ,}\tfrac{1}{\sqrt[{3}]{{x}}}\text{ )}$ over one complete transversal of the above curve.

_____________________

a)

$x=a\text{ }\cos ^{3}t$
$y=a\text{ }\sin ^{3}t$
$0\text{ }\leq t\leq \text{ }2\pi$

b) Since the field is not conservative, we have to find the line integral (i.e. the answer is not zero)... My final answer is $-6\pi a^{\tfrac{2}{3}}$

However, I think the negative sign is not correct because the force field is along the path of motion, meaning positive would make more sense...

What do you think?

The magnitude of your answer is correct. I don't know why you get a negative because I get $3 a^{2/3} \int_{t = 0}^{t = 2 \pi} \cos^2 (t) + \sin^2 (t) \, dt = 3 a^{2/3} (2 \pi - 0) = +6 \pi a^{2/3}$.