i need to find the length of the curve r(t) = i + t^2 j + t^3 k from 0 <= t <= 1 . I'm trying to follow this example here
but when I let t = e^u and replace the t in with e^u in the domain and take the natural log of all 3 parts i get math error on ln 0 and ln 1 = 0 so now I'm lost and freaking out and my test is tomorrow on chapter 12 up to this section and I realize I am probably not the best at substitution, i looked up some videos but its hard to follow when its not very well explained and its hard to transfer those examples to this. so I just want to know the best way to approach this problem. please and thanks so much.
$r'(t)=(0,2t,3t^2)$
You don't need to change parametrization!
$$L=\int_0^1 \sqrt{4t^2+9t^4}\,dt=\dfrac{1}{27}\left( 13 \sqrt{13}-8\right)\approx 1.44$$
The indefinite integral is quite easy. As $t\ge 0$ you have
$\sqrt{4t^2+9t^4}=t\sqrt{4+9t^2}=\dfrac{1}{18}\,18t\sqrt{4+9t^2}$
$=\dfrac{1}{18}\cdot \dfrac{2}{3}(4+9t^2)^{3/2}+C=\dfrac{1}{27}\,(4+9t^2)^{3/2}+C$
Plug $t=1$ and get $\dfrac{1}{27}\, 13^{3/2}$
plug $t=0$ and get $\dfrac{1}{27}\,4^{3/2}$
$L=\dfrac{1}{27}(13\sqrt{13}-8)$
Hope this helps