### Instruction: Find the length of the parametric curve described by
\[(x,y) = (2 \sin t, 2 \cos t)\]from $t = 0$ to $t = \pi.$.### Response: Well, the length of a parametric curve is just the integral of the speed.
And the speed is just the square root of the sum of the squares of the derivatives of the components.
Right. So the speed is $\sqrt{(2 \cos t)^2 + (-2 \sin t)^2} = \sqrt{4 \cos^2 t + 4 \sin^2 t}$.
But $\sin^2 t + \cos^2 t = 1$.
Exactly. So the speed is $\sqrt{4} = 2$.
So we need to find the integral of $2$ from $0$ to $\pi$.