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