Question

Find the length of the spiral r=a(Theta) on [0,Pi]

Answer 1

Do you mean r = a * (theta) ?

For each change in theta by amount dtheta, the radial coordinate increases by a*dtheta and the normal component increases by a*theta*dtheta. Using the Pythagorean theorem, the spiral length increases by
sqrt[a^2(theta^2+1)]dtheta
= a sqrt(theta^2 +1)dtheta
Integrate that from theta = 0 to pi.