length of a curve question mmn 16 1B

**transgalactic** · May 29th 2011, 12:41 AM

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

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

but what is the range of t?

**mr fantastic** · May 29th 2011, 02:23 AM

Originally Posted by transgalactic

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

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^(2/3) + y^(2/3) = 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.

**transgalactic** · June 7th 2011, 01:44 PM

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
?

**mr fantastic** · June 7th 2011, 04:11 PM

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?

**transgalactic** · June 7th 2011, 08:56 PM

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

**mr fantastic** · June 8th 2011, 03:30 AM

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^(2/3) + y^(2/3) = 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?

**transgalactic** · June 8th 2011, 03:46 AM

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

but its not an integral

??

**mr fantastic** · June 8th 2011, 03:49 AM

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.

**transgalactic** · June 8th 2011, 04:37 AM

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

i dont have it here
?

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

i ll try it

**HallsofIvy** · June 8th 2011, 06:43 AM

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
$\int \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2} dt$

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 $x= a(cos t)^3= 0$ ? For what t is $a(cos t)^3= a$ ?

**transgalactic** · June 8th 2011, 07:05 AM

t is from 0 till pi/2

and i multiply the result by 4
i understand now
thanks

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