How to find the length of the curve?

Question

#f(x) = 2sqrtx^3 #, for #0<= x <= 8/3#

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Answer 1 · 2018-04-15T16:53:30

The arc length is #9.19# units.

Recall the formula for arc length,

#A = int_a^b sqrt(1 + (dy/dx)^2)dx#

the derivative of #f(x) = 2x^(3/2)# is #f'(x) = 3x^(1/2)#.

#A = int_0^(8/3) sqrt(1 + (3x^(1/2))^2) dx#

#A = int_0^(8/3) sqrt(1 + 9x) dx#

Let #u = 9x +1#. Then #du = 9 dx# and #dx= 1/9du#. We change the bounds of integration accordingly.

#A = 1/9int_1^25 sqrt(u) du#

#A = 1/9[2/3u^(3/2)]_1^25#

#A = 1/9(2/3(25)^(3/2)) - 1/9(2/3(1)^(3/2))#

#A = 250/27 - 2/27#

#A = 248/27 ~~ 9.19#

Hopefully this helps!