calculus with parametric curves

Question

Find the distance traveled by a particle with position (x, y) as t varies in the given time interval.

x = 5 sin²(t),    y = 5 cos²(t),    0 ≤ t ≤ 5𝜋

Find the length L of the curve.

Answer 1

Consider informally the curve followed by the particle.

Notice that

x + y = 5sin²(t) + 5cos²(t) = 5(sin²(t) + cos²(t)) = 5

That is, the particle follows a straigt line.

At time 0, the particle is in position (0, 5)

then it follows a straight line to position (5, 0) reached when t = π/2.

Then it follows the straight line back to (0, 5) reached at time π.

The length of the line between (0, 5) and (5, 0) is 5√2.

And the particle traverses that a total of 10 times till the time 5π.

Therefore it travels the total distance L = 50√2.

This can also be done more formally, in case this is supposed to be an exercise in line integrals.