Exercise · ZW-GREENS-THEOREM-009 · ★★★ Advanced
Let \(C\) be the counterclockwise boundary of a region \(D\). Which of these line integrals over \(C\) gives the area of \(D\)?
Choose one option.
Show answer
- B. \(\oint_C x\,dy\)
Explanation
With \(P=0\), \(Q=x\) we get \(Q_x-P_y=1\), so \(\oint_C x\,dy=\iint_D dA\). The option \(\oint_C y\,dx\) gives the negative of the area.
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
- ZW-GREENS-THEOREM-001 Let C be the circle x^2+y^2=4 traversed counterclockwise. Compute oint_C -y dx+x dy.
- ZW-GREENS-THEOREM-002 Let C be the boundary of the rectangle [0,3]×[0,2], traversed counterclockwise. Compute oint_C y dx.
- ZW-GREENS-THEOREM-003 Compute oint_C xy dx+x^2 dy, where C is the counterclockwise boundary of the square [0,1]×[0,1].
- ZW-GREENS-THEOREM-004 Compute oint_C x^2 dx+xy dy, where C is the counterclockwise boundary of the triangle with vertices…
- ZW-GREENS-THEOREM-005 Compute oint_C y^3 dx+x^3 dy over the unit circle, counterclockwise.
- ZW-GREENS-THEOREM-006 Let C be the ellipse tfracx^24+tfracy^29=1, counterclockwise. Compute oint_C -tfrac y2 dx+tfrac x2…
- ZW-GREENS-THEOREM-007 If oint_C P dx+Q dy=5 with C counterclockwise, what is the integral over the same curve traversed c…
- ZW-GREENS-THEOREM-008 Let C be any simple closed piecewise smooth curve. What is oint_C 2xy dx+x^2 dy?