How do you find the arc length of the curve #y = 4 ln((x/4)^(2) - 1)# from [7,8]?

1 Answer
Jul 17, 2015

The final answer can be seen here.

The general formula for the arc length is as follows:

#D(x) = sqrt((Deltax)^2 + (Deltay)^2)#

#s = D(x) = sum_a^b sqrt((Deltax)^2 + (Deltay)^2/(Deltax)^2*(Deltax)^2)#

# = sum_a^b sqrt(1 + ((Deltay)/(Deltax))^2)Deltax#

# = int_a^b sqrt(1 + ((dy)/(dx))^2)dx#

Thus, take the derivative and simplify.

#(dy)/(dx) = 4*(1/((x/4)^2 - 1))*(2(x/4)*1/4)#

#= (x/((x/4)^2 - 1))*1/2#

#= x/(2(x/4)^2 - 2)#

#= x/(x^2/8 - 2)#

#= (8x)/(x^2 - 16)#

Now, plug it in and square it:
#s = int_7^8 sqrt(1 + ((8x)/(x^2 - 16))^2)dx#

#= int_7^8 sqrt(1 + (64x^2)/(x^2 - 16)^2)dx#

#= int_7^8 sqrt(((x^2 - 16)^2 + 64x^2)/(x^2 - 16)^2)dx#

If we multiply this out, we should find that something like #-32x^2# adds with the #64x^2# for a nice and sneaky shift into a perfect square.

#(x^2 - 16)^2 = x^4 - 32x^2 + 256#

#(x^2 - 16)^2 + 64x^2 = x^4 + 32x^2 + 256#

#= (x^2 + 16)^2#

Much better. Now we can get rid of that ugly square root.

#= int_7^8 sqrt(((x^2+16)^2)/(x^2 - 16)^2)dx#

#= int_7^8 (x^2+16)/(x^2 - 16)dx#

Then some manipulation to make this evaluation easier...ish.

#= int_7^8 (x^2+16 - 16 + 16)/(x^2 - 16)dx#

#= int_7^8 (x^2 - 16 + 32)/(x^2 - 16)dx#

#= int_7^8 dx + int_7^8 32/((x+4)(x-4))dx#

Looks like we probably have to do Partial Fraction Decomposition on this, unfortunately. Oh well.

#int 32/((x+4)(x-4)) = A/(x+4) + B/(x-4)#

#= (A(x-4) + B(x+4))/((x+4)(x-4))#

#= (Ax-4A + Bx+4B)/((x+4)(x-4))#

#= ((A+B)x + (-4A + 4B))/((x+4)(x-4))#

Thus, equating it back to the original equation:

#A+B = 0#
#A = -B#

#-4A + 4B = 32#
#-A + B = 8#
#2B = 8#
#B = 4 -> A = -4#

Not too bad, actually. Now we have, overall:

#= int_7^8 dx + (-int_7^8 4/(x+4)dx + int4/(x-4)dx)#

#= int_7^8 dx - int_7^8 4/(x+4)dx + int4/(x-4)dx#

#= [x]|_(7)^(8)# #- [4ln|x+4|]|_(7)^(8)# #+ [4ln|x-4|]|_(7)^(8)#

#= (8-7) - (4ln12 - 4ln11) + (4ln4 - 4ln3)#

#~~ 1.8027 "u"#

The exact answer is:
#color(blue)(1 - 4(ln12 - ln11 - ln4 + ln3))#