What is the arc length of the curve given by #y = ln(x)/2 - x^2/4 # in the interval #x in [2,4]#?
1 Answer
Mar 7, 2018
The arc length is
Explanation:
#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#