2 Finding length of curve problems

[Sympathy]

1. Find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative.

y = the integral from -2 to x of the SQUARE ROOT (3t^4-1)dt, -2 < x < -1

note that the "<" is actually less than or equal to, don't know how to post that.

For this one, do I just plug the x in? x_x I'm really clueless on how to start

2. Find the length of the curve.

y = the integral of 0 to x of SQUARE ROOT (cos(2t))dt from x = 0 to x = pi/4The problem with me is I know how to do it in terms of y and x, but I am terrible at parametrics.

If you can, please help me with any of the above =) thanks

 

[Gib Z]

Heres the arc length formula:

$$\int_a^b \sqrt{1+\frac{dy}{dx}} dx$$. Sub in the requirements, easy enough to get.

 

[Gib Z]

O sorry didnt actually read it through well. For the first one,

$$\int^{-2}_{x} \sqrt{3t^4 -1} dt$$. If there wasn't an X there, but instead a normal number like you normally see, you would find the integral and then sub in b into it, and - the integral with a subbed in. In this case just sub in X.

 

[Gib Z]

For the second one, there should be a second parametric equation >.<

 

[Sympathy]

how do you actually type in the integral sign and stuff?

 

[drpizza]

Here's one source:
http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideCommands.php
which will help with formatting the integrals

And another source for starters: (crash course in LaTeX at these forums:
https://www.physicsforums.com/misc/howtolatex.pdf

 

[drpizza]

Oh, and to see specifically how Gib Z did it, click to quote him, and take a look at what he has.

However, the formula has a small mistake in it...
(So, I copied and pasted from the quote so I could change it more simply)
$$\int_a^b \sqrt{1+(\frac{dy}{dx})^2} dx$$.
There's supposed to be a squared in there...

 

[Gib Z]

Yes, of course, drpizza's correct...I forget the squared, my bad :p

 

[Gib Z]

Btw, rather than actually having to quote me, just click on my latex pictures, that'll show up what I typed to show that code.