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I am test my knowledge of differential forms and obviously I am missing something because I can't figure out where I am going wrong here:

Let ##C## denote the positively oriented half-circle of radius ##r## parametrized by ##(x,y) = (r \cos t, r \sin t)## for ##t \in (0, \pi)##. The value of ##\int_C dx## should give the net change in ##x## as we travel from the beginning to the end of the circle, right? This change should be ##-2r## since ##C## is of radius ##r##. However, ## x = r \cos t \implies dx = -r \sin t dt \implies \int_C dx = \int_0^\pi -rsint dt = -2r^2 \neq -2r##.

Where am I going wrong?

Let ##C## denote the positively oriented half-circle of radius ##r## parametrized by ##(x,y) = (r \cos t, r \sin t)## for ##t \in (0, \pi)##. The value of ##\int_C dx## should give the net change in ##x## as we travel from the beginning to the end of the circle, right? This change should be ##-2r## since ##C## is of radius ##r##. However, ## x = r \cos t \implies dx = -r \sin t dt \implies \int_C dx = \int_0^\pi -rsint dt = -2r^2 \neq -2r##.

Where am I going wrong?

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