Using Green's Theorem, the line integral $\int $C $e ^ { x + y } d x + e ^ { x + y } d y$ where C is $x ^ { 2 } + y ^ { 2 } = 4$ is

A)3
B)2
C)1
D)0
E)-1

Question

Using Green's Theorem, the line integral $\int $C $e ^ { x + y } d x + e ^ { x + y } d y$ where C is $x ^ { 2 } + y ^ { 2 } = 4$ is

A)3
B)2
C)1
D)0
E)-1

Quizplus · Accepted Answer

Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. The given line integral is $\int_C e^{x+y} dx + e^{x+y} dy$. According to Green's Theorem, this can be converted to a double integral over the region D bounded by C: $\int \int_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA$, where $P = e^{x+y}$ and $Q = e^{x+y}$. The partial derivatives $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$ are both $e^{x+y}$, so their difference is 0. Therefore, the double integral over D of 0 is 0, which means the original line integral is also 0.

Using Green's Theorem, the Line Integral ∫\int ∫ C ex+ydx+ex+ydye ^ { x + y } d x + e ^ { x + y } d yex+ydx+ex+ydy

Using Green's Theorem, the Line Integral $\int$ C $e ^ { x + y } d x + e ^ { x + y } d y$