# Trouble calculating line integral using Green's theorem, complicated integral.

"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?

• is it y(e^x) (y a function of e^x) or ye^x? – Picaud Vincent Dec 2 '18 at 13:16
• the latter, y * e^x – MinYoung Kim Dec 3 '18 at 8:23

"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: $$\oint_C Ldx+Mdy=\int\int_D e^x-e^xdxdy=0$$