Calc. II

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For calculating the arc length of curve `y=f(x)` you  use the following formula


Now we first calculate `y'`




Now we insert that into our formula.





Now we solve each integral separately.






Now we use partial fractions.

`2int_(e^2)^(e^4)(1/(4(t-4))-1/(4t))dt=` `1/2(ln(t-4)-lnt)|_(e^2)^(e^4)=` `1/2(ln(e^4-4)-4-ln(e^2-4)+2)=` `1/2ln((e^4-4)/(e^2-4))-1`  

Now to obtain final result we just add `I_1` and `I_2.`


` `  `ln((e^4-4)/(e^2-4))-1=`


The arc length of the curve is `ln((e^4-4)/(e^3-4e))`

Sketch of the curve is given in the image below.

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