Find the length of the curve C which is parametrized by r(t) = (2 cos t, 2 sin t, √5 t) for0 ≤ t ≤ 2

To find the length of the curve C, we need to use the formula for the length of a parametric curve:

L = ∫ from a to b ||r'(t)|| dt

First, we need to find r'(t). The derivative of r(t) = (2 cos t, 2 sin t, √5 t) is r'(t) = (-2 sin t, 2 cos t, √5).

Next, we find the magnitude of r'(t):

||r'(t)|| = sqrt[(-2 sin t)^2 + (2 cos t)^2 + (√5)^2] = sqrt[4 sin^2 t + 4 cos^2 t + 5] = sqrt[4(sin^2 t + cos^2 t) + 5] = sqrt[4*1 + 5] = sqrt[9] = 3

Now, we can find the length of the curve C by integrating ||r'(t)|| from 0 to 2:

L = ∫ from 0 to 2 ||r'(t)|| dt = ∫ from 0 to 2 3 dt = [3t] from 0 to 2 = 32 - 30 = 6

So, the length of the curve C is 6.