Question

1.) Find the length of the arc of the graph x^4 = y^6 from x = 1 to x = 8.

2.) Find the length of arc with parametric equations x = 3 cos t, y = 3 sin t from t=0 to t=⛩

Answer 1

I believe I've addressed (1) in another question of yours (24529718).

For (2), the arc length of the curve parameterized by x(t) = 3 cos(t ) and y(t) = 3 sin(t ) over 0 ≤ t ≤ π is

`\displaystyle \int_0^\pi \sqrt{\left((\mathrm dx)/(\mathrm dt)\right)^2 + \left((\mathrm dy)/(\mathrm dt)\right)^2}\,\mathrm dt`

We have

`(\mathrm dx)/(\mathrm dt) = -3\sin(t) \text{ and }(\mathrm dy)/(\mathrm dt) = 3\cos(t)`

so that the integral reduces to

`\displaystyle \int_0^\pi √(9\sin^2(t) + 9\cos^2(t))\,\mathrm dt = 3\int_0^\pi\mathrm dt`

since
`\cos^2(t)+\sin^2(t)=1` for all t. The remaining integral is trivial:

`\displaystyle 3\int_0^\pi\mathrm dt = 3t\bigg|_0^\pi = 3(\pi-0) = \boxed{3\pi}`