Use Green's Theorem to evaluate the line integral $\int _ { C } e ^ { x } \cos ( 13 y ) d x - 13 e ^ { x } \sin ( 13 y ) d y$ where $C$ is $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$
A) 13
B) 14
C) 170
D) 26
E) 0

Question

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 theorem states that if $L$ and $M$ are functions of $(x, y)$ defined on an open region containing $D$ and $C$, and if $L_x$ and $M_y$ are their partial derivatives that exist and are continuous, then:$$\oint_{C} (L dx + M dy) = \iint_{D} \left( \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \right) dA$$For the given integral, $L = e^x \cos(13y)$ and $M = -13e^x \sin(13y)$. Thus, we find the partial derivatives:$$\frac{\partial M}{\partial x} = -13e^x \sin(13y)$$$$\frac{\partial L}{\partial y} = -13e^x \sin(13y)$$Substituting these into the formula gives:$$\iint_{D} \left( -13e^x \sin(13y) - (-13e^x \sin(13y)) \right) dA = \iint_{D} 0 dA$$The integral of 0 over any region is 0, hence the answer is E.

Use Green's Theorem to Evaluate the Line Integral ∫Cexcos⁡(13y)dx−13exsin⁡(13y)dy\int _ { C } e ^ { x } \cos ( 13 y ) d x - 13 e ^ { x } \sin ( 13 y ) d y∫C​excos(13y)dx−13exsin(13y)dy

Use Green's Theorem to Evaluate the Line Integral $\int _ { C } e ^ { x } \cos ( 13 y ) d x - 13 e ^ { x } \sin ( 13 y ) d y$