Exercise · ZW-GREENS-THEOREM-001 · ★☆☆ Basic
Let \(C\) be the circle \(x^2+y^2=4\) traversed counterclockwise. Compute \(\oint_C -y\,dx+x\,dy\).
Choose one option.
Show answer
- B. \(8\pi\)
Explanation
Here \(Q_x-P_y=1-(-1)=2\), so the integral is \(2\) times the area of the disk: \(2\cdot4\pi=8\pi\).
Review the concept: Green's Theorem
Green's theorem equates the circulation of a field \(\mathbf F=\langle P,Q\rangle\) along a simple closed curve \(C\), counterclockwise, with a double integral over its region \(D\), with respect to area \(A\).
\[\oint_C P\,dx+Q\,dy=\iint_D\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA\]
More practice problems: green's theorem
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