r(t) = cos^3 t i + sin^3 t j; 0 < t < pi/2.
r'(t) = -3cos^2 t sin t + 3 sin^2 t cos t
||r'(t)|| = 3 sin t cos t
Now to find the arc length parameterization, we need S = integration from t0 to t of ||dr/du|| du.
Question: What is S here?
Which gives 3/2 sin^2 t.
After finding that, the book writes these stuff:
What do these things mean?
To find the arc length parameterization of a 3d curve, you should follow the following steps:
1) Find the arc length function
2) Solve for the function in step 1 for $t$
3) Replace $t$ with what you found in step 2
The arc length function is $$s(t) = \int_0^t |r'(u)|\hspace{1mm}du$$
$$s(t) = \int_0^t 3\sin u\cos u\hspace{1mm}du$$
$$s = \bigg[ \dfrac{3}{2}\sin^2 u\bigg]_0^t$$
$$s = \dfrac{3}{2}\sin^2 t$$
Going into step 2: Now we solve for $t$
$$ \dfrac{2s}{3}=\sin^2 t$$
Take square root
$$ \sqrt{\dfrac{2s}{3}}=\sin t$$
$$ \arcsin\sqrt{\dfrac{2s}{3}}= t$$
Step 3: Substitute this in the given parametrization
$$r(s) = \left(\sqrt{1-\dfrac{2s}{3}}\right)^3i+\left(\sqrt{\dfrac{2s}{3}}\right)^3j$$