Find the length of the curve defined by y 5 Inx52 1 from x

Find the length of the curve defined by y = 5 In(x/5)^2 - 1) from x = 7 to x = 8.

Solution

The formula to find arc length is :

L = a to b (1 + (y\')^2) dx

We have y = 5ln((x/5)^2 - 1) and a = 7, b = 8

y = 5ln[((x/5) - 1)*((x/5) + 1)] …[by (a+b)(a-b) = a^2 – b^2]

= 5ln((x/5) - 1) + 5ln((x/5) + 1) ….[by ln (a*b) = ln a + ln b]

= 5*[ln((x - 5)/5) + ln((x + 5)/5)]

Now differentiating w.r.t x, we get

y\' = 5*(1/5) [1/((x - 5)/5) + 1/((x + 5)/5)]

= 5/(x - 5) + 5/(x + 5)
= 10x/(x^2-25)

Now (y’)^2 = 100x^2/(x^2-25)^2
And 1 + (y\')^2 = 1 + 100x^2/(x^2-25)^2
= (x^2+25)^2/(x^2-25)^2

Now (1 + (y\')^2) = (x^2+25)/(x^2-25)
= (x^2 - 25 + 50)/(x^2 - 25)

= 1 + 50/(x^2 - 25)

Applying partial fractions method to get :

50/(x^2 - 25) = 5/(x - 5) - 5/(x + 5)

Now finally,
L = a to b (1 + (y\')^2) dx
L = 7 to 8 [(1 + 5/(x - 5) - 5/(x + 5) ] dx

= x + 5*ln|(x-5)/(x+5)| (7 to 8)
= 8 + 5*ln(3/13) - 7 - 5*ln(2/12) = 1 + 5*ln(36/26)
= 1 + 5*0.32542
= 2.62711