Question

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!

Answer 1

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 `2ax+b=u`. Let `c=50.4a+b`:

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

Apply the substitution `u=tantheta`:

`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 `a`, `b`, and `c` for the final answer.