"Calculate line integral of scalar function [$y(e^x) -1]dx + [e^x]dy$ over curve $C$, where $C$ is the semicircle through $(0, 10), (10, 0)$, and $(0, 10)$"
I plan on using Green's theorem, and since curve C does not include the bottom line on x axis, subtract line integral over this line on x axis from result of area integral from Green's theorem. However, I am having trouble calculating the line integral over this bottom curve, I've tried using cartesian and polar coordinates, but it gets very complex. Am I doing something wrong?
Best Answer
Green's theorem states that: wiki
"for a curve C positively oriented, piecewise smooth, simple closed in a plane, and D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives", then: $$ \oint_C Ldx+Mdy=\int\int_D \frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}dxdy $$ Your example: $L=ye^x-1$, $M=e^x$ Thus: $$ \oint_C Ldx+Mdy=\int\int_D e^x-e^xdxdy=0 $$