Integration problem regarding parabolic length

Limits, differentiation, related rates, integration, trig integrals, etc.

Integration problem regarding parabolic length

$\mbox{Hence, the total length of the cable can be}$

$\mbox{determined by integration. Using Eqn. 4, we have}$

. . . . .$\mathscr{L}\, =\, \displaystyle{\int\, ds\, =\, 2\, \int_{0}^{L/2}\, \sqrt{1\, +\, \left(\frac{8h}{L^2}x\right)^2\,}\, dx}$

$\mbox{Integrating yields}$

. . . . .$\mathscr{L}\, =\, \displaystyle{\frac{L}{2}\, \left[\,\sqrt{1\, +\, \left(\frac{4h}{L}\right)^2\, }\, +\, \frac{L}{4h}\sinh^{-1}\left(\frac{4h}{L}\right)\,\right]}$

How did the author arrive at his answer? I've been racking my brain around this for days to no avail... :(
mightytosave

Posts: 3
Joined: Tue Jun 03, 2014 11:55 pm

mightytosave wrote:
$\mbox{Hence, the total length of the cable can be}$

$\mbox{determined by integration. Using Eqn. 4, we have}$

. . . . .$\mathscr{L}\, =\, \displaystyle{\int\, ds\, =\, 2\, \int_{0}^{L/2}\, \sqrt{1\, +\, \left(\frac{8h}{L^2}x\right)^2\,}\, dx}$

$\mbox{Integrating yields}$

. . . . .$\mathscr{L}\, =\, \displaystyle{\frac{L}{2}\, \left[\,\sqrt{1\, +\, \left(\frac{4h}{L}\right)^2\, }\, +\, \frac{L}{4h}\sinh^{-1}\left(\frac{4h}{L}\right)\,\right]}$

How did the author arrive at his answer? I've been racking my brain around this for days to no avail... :(

What is "Eqn. 4"? What preceded the exercise (such as "the cable" in question)?

Thank you!

stapel_eliz

Posts: 1716
Joined: Mon Dec 08, 2008 4:22 pm

Re: Integration problem regarding parabolic length

Thanks for the reply! Eqn. 4 is 8hx/L^2.
mightytosave

Posts: 3
Joined: Tue Jun 03, 2014 11:55 pm

mightytosave wrote:Eqn. 4 is 8hx/L^2.

Lacking an "equals" sign, this is just an expression, not an equation. What was on the other side of the "equals" sign? How are the various variables defined? Thank you!

stapel_eliz

Posts: 1716
Joined: Mon Dec 08, 2008 4:22 pm

Re: Integration problem regarding parabolic length

Here is the whole problem. I got lost when integrating eq. (7) on page 2. Thank you for replies! really appreciate them.
mightytosave

Posts: 3
Joined: Tue Jun 03, 2014 11:55 pm