How do you calculate the arc length of the curve #y=x^2# from #x=0# to #x=4#?

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Answer 1 · 2018-05-16T06:33:34

Use the arc length formula.

#y=x^2#

#y'=2x#

Arc length is given by:

#L=int_0^4sqrt(1+4x^2)dx#

Apply the substitution #2x=tantheta#:

#L=1/2intsec^3thetad theta#

This is a known integral:

#L=1/4[secthetatantheta+ln|sectheta+tantheta|]#

Reverse the substitution:

#L=1/4[2xsqrt(1+4x^2)+ln|2x+sqrt(1+4x^2)|]_0^4#

Hence

#L=2sqrt65+1/4ln(8+sqrt65)#