Line integral?

Question

∫c (sinx+cosy)ds, where c is the line segment from (0,0) to (pi, 2pi)

Answer 1

Parameterize as follows:

x(t) = πt
y(t) = 2πt

Then the integral is:

[c]∫sin (πt) + cos (2πt) ds

ds=√((dx/dt)² + (dy/dt)²) dt

dx/dt = π
dy/dt = 2π

ds=√(π² + 4π²) dt = π√5 dt

[0, 1]∫π√5 (sin (πt) + cos (2πt)) dt

This you should be able to integrate. The answer is 2√5

Edit (re: james k) - your answer uses the implicit parameterization x(t) = t, y(t) = 2t. Either one will work, provided that ds is computed correctly. However, in your case, you forgot to compute it at all (it turns out to be √5 dt, so the correct answer with your method is 2√5, which agrees with mine).

Answer 2

I think that the last person had it mostly right but what you need, I believe, is the integral between t=0..Pi of (sin(t)+cos(2t)). This works out to be [-cos(t)+sin(2t)/2]pi...0= 2