What is the arc length of the curve given by #y = ln(x)/2 - x^2/4 # in the interval #x in [2,4]#?

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Answer 1 · 2018-03-07T16:58:01

The arc length is #1/2ln2+3# units.

#y=1/2lnx-1/4x^2#

#y'=1/2(1/x-x)#

Arc length is given by:

#L=int_2^4sqrt(1+1/4(1/x-x)^2)dx#

Factor out the constant and expand:

#L=1/2int_2^4sqrt(4+(1/x^2-2+x^2))dx#

Simplify:

#L=1/2int_2^4sqrt(1/x^2+2+x^2)dx#

Factorize:

#L=1/2int_2^4sqrt((1/x+x)^2)dx#

Simplify:

#L=1/2int_2^4(1/x+x)dx#

Integrate term by term:

#L=1/2[lnx+1/2x^2]_2^4#

Insert the limits of integration:

#L=1/2ln2+3#