Question

Find the length of the arc with the curve

y = 2x ^ (3/2) between x = 1/3 and x = 7

Answer 1

Length of the arc: $L= \int_a^b \sqrt{1+(y')^2}dx$

$y = 2x^{3/2}$

$y' = 3x^{1/2}$

$(y')^2 = 9x$

$L = \int_{1/3}^7 \sqrt{1+9x}dx =[t = 1+9x, dt=9dx, dx= \cfrac{dt}{9}, \cfrac{1}{3} \rightarrow4, 7 \rightarrow 64] =\cfrac{1}{9} \int_4^{64}\sqrt{t} dt =\cfrac{2}{27} t^{3/2}|_4^{64} = \cfrac{2}{27}(64^{3/2} - 4^{3/2}) = \cfrac{2}{27}(512-8) =\cfrac{2}{3}56 = \cfrac{112}{3}$