Find the length of the spiraling polar curve?

Question

r=4e^2@, from 0 to 2pi

Answer 1

I assume you mean r = 4e^(2θ). First, we parameterize this equation in terms of t and switch to rectangular coordinates. Let θ = t and r = 4e^(2t), so that x = r cos θ = 4e^(2t) cos t and y=r sin θ = 4e^(2t) sin t. Then we have that:

dx/dt = 8e^(2t) cos t - 4e^(2t) sin t
dy/dt = 8e^(2t) sin t + 4e^(2t) cos t

(dx/dt)² = 64e^(4t) cos² t - 64 e^(4t) cos t sin t + 16 e^(4t) sin² t = 16e^(4t) (4 cos² t - 4 cos t sin t + sin² t)

(dy/dt)² = 64e^(4t) sin² t + 64 e^(4t) cos t sin t + 16e^(4t) cos² t = 16e^(4t) (4 sin² t + 4 cos t sin t + cos² t)

Now, to find the arc length of the curve, we just plug the numbers into the formula:

[0, 2π]∫√((dx/dt)² + (dy/dt)²) dt

[0, 2π]∫√(16e^(4t) (4 cos² t - 4 cos t sin t + sin² t) + 16e^(4t) (4 sin² t + 4 cos t sin t + cos² t)) dt

[0, 2π]∫4e^(2t) √(5 cos² t + 5 sin² t) dt

[0, 2π]∫4e^(2t) √5 dt

2√5 e^(2t) |[0, 2π]

2√5 e^(4π) - 2√5

And we are done.