Find Arc Length of Astroid: Integral Solution

[whatlifeforme]

Homework Statement

the graph of the equation x^(2/3) + y^(2/3) = 7^(2/3) is one of the family of curves called asteroids.

Homework Equations

find the length of first-quadrant and multiply by 8.
1. y=(7^(2/3) - x^(2/3))^(3/2)) ; 7sqrt(2)/4 <= x <= 7

The Attempt at a Solution

1. i found dy/dx, (dy/dx)^2, and now I'm at the length integral.
2. L = integral sqrt(1+7^(2/3) * x^(2/3) - x^(4/3))

i'm not sure how to evaluate this integral (#2) without using a calculator, but that doesn't give me the exact answer that i need.

 

[LCKurtz]

I get a simpler expression for ##1 + y'^2##. Please show your work.

 

[whatlifeforme]

dy/dx = -x^1/3 (7^2/3 - x^2/3)^(1/2)
(dy/dx)^2 = x^(2/3) * (7^(2/3) - X^(2/3)V ----> 7^(2/3)* X^(2/3) - X^(4/3)

L = ∫ (x=a to x=b) SQRT(1 + 7^(2/3) * x^(2/3) - x^(4/3))

 

[haruspex]

Instead of converting the equation to the form y = f(x), just differentiate the equation as given. It's less messy, so less prone to error.

 

[whatlifeforme]

it's already solved for y that's how the want you to solve it.

 

[LCKurtz]

it's already solved for y that's how the want you to solve it.

That is not the easy way to solve it and is probably why you are having trouble. Differentiate the equation ##x^{2/3} + y^{2/3} = 7^{2/3}## implicitly with respect to ##x## to get ##y'## and calculate ##1 + y'^2##. Show us what you get.