# How could I find the arc length of the following function: y= -0.061016737619069x^2 + 4.3435741689529x {0≤x≤ 25.20}?

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Plz can you show me all of the required steps for me to do this. I completely do not understand this. Thank you!

Plz can you show me all of the required steps for me to do this. I completely do not understand this. Thank you!

##### 1 Answer

Apply the arc length formula and solve it using trigonometric substitution.

#### Explanation:

Given

#y=-0.061016737619069x^2+4.3435741689529x#

For clarity, let:

#a=-0.061016737619069# and#b=4.3435741689529#

Hence

#y=ax^2+bx#

#y'=2ax+b#

#(y')^2=(2ax+b)^2#

Arc length is given by the formula:

#L=intsqrt(1+(y')^2)dx#

Hence

#L=int_0^25.2sqrt(1+(2ax+b)^2)dx#

Apply the substitution

#L=1/(2a)int_b^csqrt(1+u^2)du#

Apply the substitution

#L=1/(2a)intsec^3thetad theta#

This is a known integral. If you do not have it memorized apply integration by parts or look it up in a table of integrals:

#L=1/(4a)[secthetatantheta+ln|sectheta+tantheta|]#

Reverse the last substitution:

#L=1/(4a)[usqrt(1+u^2)+ln|u+sqrt(1+u^2)|]_b^c#

Hence

#L=1/(4a)[csqrt(1+c^2)-bsqrt(1+b^2)+ln|(c+sqrt(1+c^2))/(b+sqrt(1+b^2))|]#

Insert the values of