1. ## Length of curve

$\displaystyle y=3x^3-9x^2$
Domain: $\displaystyle [0,3]$

What's the lenght of curve ?

2. Hello, totalnewbie!

I can set it up, but I don't see a way to integrate it . . .
. . Is there a typo?

$\displaystyle y \:=\:3x^3-9x^2$
Domain: $\displaystyle [0,3]$
What's the length of curve?

Formula: .$\displaystyle L \;= \;\int^b_a\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$

We have: .$\displaystyle \frac{dy}{dx} \:=\:9x^2 - 18x$

Hence: .$\displaystyle L \;=\;\int^3_0\sqrt{1 + (9x^2 - 18x)^2}\:dx \;= \;\int\sqrt{81x^4 - 324x^3 + 324x^2 + 1}\:dx$

. . Good luck!

3. When you have,
$\displaystyle \int \sqrt{P(x)}dx$
Where $\displaystyle P(x)$ is a cubic or quartic polynomial. Usually the integral leads to an "elliptic integral" and fails to be integrated.
----
Using Soroban's integral.
And Simpson's Rule with 100 terms
I got,
$\displaystyle \approx 24.3662$

4. Originally Posted by Soroban
Hello, totalnewbie!

I can set it up, but I don't see a way to integrate it . . .
. . Is there a typo?

Formula: .$\displaystyle L \;= \;\int^b_a\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\,dx$

We have: .$\displaystyle \frac{dy}{dx} \:=\:9x^2 - 18x$

Hence: .$\displaystyle L \;=\;\int^3_0\sqrt{1 + (9x^2 - 18x)^2}\:dx \;= \;\int\sqrt{81x^4 - 324x^3 + 324x^2 + 1}\:dx$

. . Good luck!

One can, of course, do it numerically either via the trapezoid or Simpson's rule, to name two of the easier methods. I did it on my calculator and got: