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

Answer 1

#sqrt13-sqrt5+2*ln((sqrt65+2*sqrt13-2*sqrt5-4))~=3.006#

To do this we need to apply the formula for the length of the curve mentioned in: How do you find the length of a curve using integration?

We start from #L=int_a^b sqrt(1+(f' (x))^2 )dx# #f(x)=lnx^2=2lnx # => #f'(x)=2/x# Then #L=int_1^3 sqrt(1+4/x^2)dx=int_1^3 sqrt(x^2+4)/xdx=F(x=3)-F(x=1)#
Making #x=2tany# #dx=2sec^2y*dy# we get #F(x)=int sqrt(x^2+4)/xdx=int(2secy*cancel2sec^2y)/(cancel2tany)dy=2int (1/cos^3y)(cosy/siny)dy=2int dy/(cos^2y*siny)#
But #1/(cos^2y*siny)=(siny)/(cos^2y)+1/siny#
So #F(x)=2int siny/cos^2ydy+2int dy/siny#
Making #cosy=z# => #siny*dy=-dz# The first part becomes #-2int dz/z^2=2/z=2/cosy#
Therefore #F(x)=2/cosy+2ln |csc y-coty| +const.# But #x=2tany# => #siny=x/2cosy# #sin^2y+cos^2y=1# => #(x^2/4+1)cos^2y=1# => #cosy=2/sqrt(x^2+4)# #-> siny=x/2.(2/sqrt(x^2+4))# => #siny=x/sqrt(x^2+4)# So #F(x)=sqrt(x^2+4)+2ln |sqrt(x^2+4)/x-2/x|+const.# #F(x)=sqrt(x^2+4)+2ln |sqrt(x^2+4)-2|-2ln|x|+const.#
Finally #L=F(x=3)-F(x=1)=sqrt13+2ln (sqrt13-2)-2ln3-(sqrt5+2ln(sqrt5-2)-2ln1)# #L=sqrt13-sqrt5+ln((sqrt13-2)/(3(sqrt5-2)))# But #(sqrt13-2)/(sqrt5-2)*(sqrt5+2)/(sqrt5+2)=sqrt65+2sqrt13-2sqrt5-4# So
#L=sqrt13-sqrt5+ln((sqrt65+2sqrt13-2sqrt5-4)/3)~=3.006#
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Answer 2

To find the arc length of the curve ( f(x) = \ln(x^2) ) on the interval ( x \in [1,3] ), we use the arc length formula:

[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} , dx ]

Where ( \frac{dy}{dx} ) represents the derivative of ( f(x) ).

First, find ( \frac{dy}{dx} ):

[ f(x) = \ln(x^2) ] [ f'(x) = \frac{d}{dx} \ln(x^2) = \frac{1}{x^2} \cdot 2x = \frac{2}{x} ]

Now, plug ( f'(x) ) into the arc length formula:

[ L = \int_{1}^{3} \sqrt{1 + \left( \frac{2}{x} \right)^2} , dx ]

[ L = \int_{1}^{3} \sqrt{1 + \frac{4}{x^2}} , dx ]

This integral can be challenging to evaluate directly, so it may be simplest to use numerical methods or a computer algebra system to approximate the value of ( L ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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