so what's the trouble? It's just a straightforward arc length problem.
y = (4 - x^(2/3))^(3/2)
y' = - √(4-x^(2/3)) / ∛x
So the arc length in QI is
s = ∫[0,8] √(1+y'^2) dx
= ∫[0,8] √(1+(- √(4-x^(2/3)) / ∛x)^2) dx
= ∫[0,8] √(1+(4-x^(2/3)) / x^(2/3)) dx
= ∫[0,8] 2/∛x dx
= 12
The equation x^(2/3) + y^(2/3) = 4 describes an asteroid. Determine the total length of this asteroid by finding the length of its portion located in the first quadrant, the equation y = (4 - x^(2/3))^(3/2) for 0≤ x ≤ 8 and multiplying this value by 4 (by symmetry).
1 answer