# What is the arc length of #f(x)= e^(3x)/x+x^2e^x # on #x in [1,2] #?

##### 1 Answer

Recall the distance formula:

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

Now extend that to arc length:

#s = D(x) = sumsqrt((Deltax)^2 + (Deltay)^2/(Deltax)^2*(Deltax)^2)#

# = sumsqrt(1 + ((Deltay)/(Deltax))^2)(Deltax)#

#=> color(blue)(s = int_a^b sqrt(1 + ((dy)/(dx))^2)dx)#

This is just a "dynamic", infinitesimally-short-distance formula that accumulates over an interval of constantly increasing

Thus, we need the first derivative of

#(dy)/(dx) = d/(dx)[e^(3x)/x + x^2e^x]#

#= e^(3x) cdot -1/x^2 + 1/x cdot 3e^(3x) + x^2 cdot e^x + e^x cdot 2x#

#= -e^(3x)/x^2 + (3e^(3x))/x + x^2e^x + 2xe^x#

#= (3x - 1)e^(3x)/x^2 + x(x + 2)e^x#

Now, in the expression, we would next square it:

#((dy)/(dx))^2 = [(3x - 1)e^(3x)/x^2 + x(x + 2)e^x]^2#

Using Wolfram Alpha, this was simplified to:

#= e^(2x)/x^4 [(x+2)x^3 + e^(2x)(3x - 1)]^2#

Inserting it into the equation for arc length, we obtain the integral that we would have attempted:

#color(green)(s = int_(1)^(2) sqrt(1 + e^(2x)/x^4 [(x+2)x^3 + e^(2x)(3x - 1)]^2)dx)#

*There is no solution for this in terms of standard mathematical functions.*

The **numerical solution** is:

#color(blue)(s = 208.471)#