# Thread: length of a curve question mmn 16 1B

1. ## length of a curve question mmn 16 1B

find the length of$\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$
?

i use the parametrisation
y=a(sint)^3
x=a(cost)^3

but what is the range of t?

2. Originally Posted by transgalactic
find the length of$\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$
?

i use the parametrisation
y=a(sint)^3
x=a(cost)^3

but what is the range of t?
In the first quadrant the curve looks like this: plot x&#94;&#40;2&#47;3&#41; &#43; y&#94;&#40;2&#47;3&#41; &#61; 1 - Wolfram|Alpha

I suggest you find the arclength for the curve in the first quadrant - the domain for t should be obvious - and then multiply by 4.

3. ok how to find the length of it

i need to use the parametrisation and make the integral over t

how to do it here
?

4. Originally Posted by transgalactic
ok how to find the length of it

i need to use the parametrisation and make the integral over t

how to do it here
?
Your original question has been answered. What work have you done, where exactly are you stuck?

5. i asked for the range of "t" in the integral

6. Originally Posted by transgalactic
i asked for the range of "t" in the integral
I posted this:

Originally Posted by Mr Fantastic
In the first quadrant the curve looks like this: plot x&#94;&#40;2&#47;3&#41; &#43; y&#94;&#40;2&#47;3&#41; &#61; 1 - Wolfram|Alpha

I suggest you find the arclength for the curve in the first quadrant - the domain for t should be obvious - and then multiply by 4.
Where are you stuck here?

7. the length of a curve is a root of the some of the partial derivatives

but its not an integral

??

8. Originally Posted by transgalactic
the length of a curve is a root of the some of the partial derivatives

but its not an integral

??
What you have posted here makes no sense. I know you understand how to find the length of a curve because of other of your threads. Show your work and say where you are stuck here.

9. this chapter is about curve integrals
integral of the form $\displaystyle \int_c Fdr$

i dont have it here
?

maybe here F=1
and we have
$\displaystyle \int_c dr$

i ll try it

10. You've changed the question so many times, it is hard to be sure what you are asking.

Surely you know that if you are writing x and y in terms of parameter t, then the arclength is given by
$\displaystyle \int \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2} dt$

That is an integral and there are no partial derivatives.

You said you wanted to use the parameterization y=a(sint)^3, x=a(cost)^3.

In the graph Mr. Fantastic showed you initially, x goes from 0 up to 1 while y goes from 1 down to 0. (He set a= 1.)
For what t is $\displaystyle x= a(cos t)^3= 0$? For what t is $\displaystyle a(cos t)^3= a$?

11. t is from 0 till pi/2

and i multiply the result by 4
i understand now
thanks