Find the length of the curve defined by #y=3ln((x/3)^2−1)# from x=7 to x=10?
1 Answer
May 25, 2018
Explanation:
#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)#