Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

Question

Use Green's Theorem to evaluate  , where C is the square bounded by x = y = 0, and x = y = 1.

Quizplus · Accepted Answer

Since the square is bounded by x=y=0 and x=y=1, we can parameterize C as x=t and y=0 from t=0 to t=1, then x=1 and y=t from t=0 to t=1, then x=t and y=1 from t=1 to t=0, and finally x=0 and y=t from t=1 to t=0. We can then use Green's Theorem:

$\oint_C (-y\ dx+x\ dy) = \iint_D \left(\frac{\partial}{\partial x}(x)-\frac{\partial}{\partial y}(-y)ight)dA=\iint_D 2\ dA=2\cdot(	ext{Area of }D)=2\cdot1=2$

Therefore, the answer is (C). However, we have to remember that the orientation of the curve matters when applying Green's Theorem. In this case, we assumed counterclockwise orientation, which is the standard convention. If the curve had been oriented clockwise, we would have gotten the opposite sign in the line integral, resulting in an answer of -2.

Use Green's Theorem to Evaluate , Where C Is