Find the length of the curve defined by #y=3ln((x/3)^2−1)# from x=7 to x=10?

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Answer 1 · 2018-05-25T20:02:53

#L=3+3ln(35/26)# units.

#y=3ln((x/3)^2−1)=3ln(x^2-9)-3ln9#

#y'=(6x)/(x^2-9)#

Arc length is given by:

#L=int_7^10sqrt(1+(36x^2)/(x^2-9)^2)dx#

Rearrange:

#L=int_7^10sqrt((x^2+9)^2)/(x^2-9)dx#

Simplify:

#L=int_7^10(x^2+9)/(x^2-9)dx#

Apply partial fraction decomposition:

#L=int_7^10(1+3/(x-3)-3/(x+3))dx#

Integrate term by term:

#L=[x+3ln|x-3|-3ln|x+3|]_7^10#

Hence

#L=3+3ln(35/26)#