What is the arc length of #f(x)=(3x)/sqrt(x-1) # on #x in [2,6] #?

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Answer 1 · 2018-07-08T18:45:29

#approx 4.514518184#

Writing

#f(x)=3x(x-1)^(-1/2)#
then by the product and the chain rule we get

#f'(x)=3(x-1)^(-1/2)+3x*(-1/2)*(x-1)^(-3/2)#

which simplifies to

#f'(x)=(3(x-2))/(2*(x-1)^(3/2))#
so we have to solve

#int_2^6 sqrt(1+(3(x-2)/(2*(x-1)^(3/2)))^2)dx#
by a numerical method we get

#\approx4.514518184#