Calculate the line integration of the vector field F= (x+y)i+(x^2+y^2)j along the unit circle clockwise.
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Using x = cos t, y = -sin t for t in [0, 2π] for clockwise orientation:
∫c · dr
= ∫(t = 0 to 2π) · <-sin t, -cos t> dt
= ∫(t = 0 to 2π) (-sin t cos t + sin^2(t) - cos t) dt
= ∫(t = 0 to 2π) (-sin t cos t + (1/2)(1 - cos(2t)) - cos t) dt
= [-sin^2(t)/2 + (1/2)(t - sin(2t)/2) - sin t] {for t = 0 to 2π}
= π.
I hope this helps!
∫c
= ∫(t = 0 to 2π)
= ∫(t = 0 to 2π) (-sin t cos t + sin^2(t) - cos t) dt
= ∫(t = 0 to 2π) (-sin t cos t + (1/2)(1 - cos(2t)) - cos t) dt
= [-sin^2(t)/2 + (1/2)(t - sin(2t)/2) - sin t] {for t = 0 to 2π}
= π.
I hope this helps!