Find the length of the curve y=(3÷4)x^(4÷3)-(3÷8)x^(2÷3)+5,1<=x<=8?

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Answer 1 · 2018-06-22T02:41:48

#L=99/8# units.

#y=3/4x^(4/3)-3/8x^(2/3)+5#

#y'=x^(1/3)-1/4x^(-1/3)#

Arc length is given by:

#L=int_1^8sqrt(1+(x^(1/3)-1/4x^(-1/3))^2)dx#

For ease of reading, apply the substitution #x=u^3#:

#L=int_1^2sqrt(1+(u-1/(4u))^2)(3u^2du)#

Simplify:

#L=3int_1^2u^2sqrt((u+1/(4u))^2)du#

Hence:

#L=3/4int_1^2(4u^3+u)du#

Integrate directly:

#L=3/4[u^4+1/2u^2]_1^2#

Insert the limits of integration:

#L=99/8#