3How do you find the lengths of the curve $y = \frac{2}{3} {(x + 2)}^{\frac{3}{2}}$ $y=2/3(x+2)^(3/2)$ for $0 \leq x \leq 3$ $0<=x<=3$ ?

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Answer 1 · 2017-02-18T15:03:05

$= 4 \sqrt{6} - 2 \sqrt{3}$ $=4 sqrt 6 - 2 sqrt 3$

Length, s, is:

$s = int_0^3 sqrt(1+ (y')^2) dx$

$y' = (x+2)^(1/2)$

$implies s = int_0^3 sqrt(3+ x) dx$

$= [2/3(3+x)^(3/2) ]_0^3$

$=2/3( 6^(3/2) - 3^(3/2) )$

$=4 sqrt 6 - 2 sqrt 3$

3How do you find the lengths of the curve y=2/3(x+2)^(3/2)y=23(x+2)32y=2/3(x+2)^(3/2) for 0<=x<=30≤x≤30<=x<=3?