Exercise · ZW-GREENS-THEOREM-004 · ★☆☆ Basic
Compute \(\oint_C x^2\,dx+xy\,dy\), where \(C\) is the counterclockwise boundary of the triangle with vertices \((0,0)\), \((1,0)\) and \((0,1)\).
Choose one option.
Show answer
- D. \(\tfrac16\)
Explanation
\(Q_x-P_y=y-0=y\) and \(\int_0^1\!\int_0^{1-x}y\,dy\,dx=\tfrac16\).
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-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?
- ZW-GREENS-THEOREM-009 Let C be the counterclockwise boundary of a region D. Which of these line integrals over C gives th…