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:

$L=1/2ln2+3$

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