Question

The formula to find the length of the curve r = f(t), y = g(t) from t = r, to t = ta is​

Answer 1

Answer:

Recall Alternative Formulas for Curvature, which states t. ... s = ∫ a b [ f ′ ( t ) ] 2 + [ g ′ ( t ) ] 2 d t = ∫ a b ‖ r ′ ( t ) ‖ d t .

Step-by-step explanation:

Answer 2

Answer:

The answer is 6√3 .

The arclength of a parametric curve can be found using the formula: L

=∫tfti√(dxdt)2+(dydt)2dt.

Since x and y are perpendicular, it's not difficult to see why this computes the arclength.

It isn't very different from the arclength of a regular function: L

=∫ba√1+(dydx)2dx

If you need the derivation of the parametric formula, please ask it as a separate question.

We find the 2 derivatives:dxdt

=3−3

t2dydt

=6t

And we substitute these into the integral:L

=∫√30√(3−3t2)2+(6t)2dt

And solve:

=∫√30√9−18t2+9t4+36t2dt

=∫√30√9+18t2+9t4dt

=∫√30√(3+3t2)2dt

=∫√30(3+3t2)dt

=3t+t3∣∣√30

=3√3+3√3

=6√3

Be aware that arclength usually has a difficult function to integrate. Most integrable functions look like the above where a binomial is squared and adding the two terms will flip the sign of the binomial.