Find the length of the curve Arc length Find the length of

Find the length of the curve

Arc length =

Solution

y = x^2/2 - lnx / 4 [2,4]

ArcLength = integral from a to b of sqrt(1 + [f \' (x)]^2 ) dx

y \' = x - 1/(4x)

integral from 2 to 4 of sqrt [ 1 + (x - 1/(4x) )^2 ] dx

int from 2 to 4 of sqrt[1 + x^2 - 1/2 + 1/(16x^2) ] dx
===>which simplifies to
x^2 + 1/2 + 1/(16x^2)
= [ x+1/(4x) ] ^2
square root of this cancels the exponent ===>

int from 2 to 4 of [ x + 1/(4x) ] dx
=x^2 / 2 +lnx /4 }eval from 2 to 4
= [4^2 / 2 + ln 4 /4 ] - [2^2 / 2 + ln 2 / 4]
= [8 + 1/4 ln4 - 2 - 1/4 ln2 ] ===> ln4 = 2ln2
= 6 + (1/4) ln 2
=6.173