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Green’s Theorem relates the value of a(n) _____ integral along a simple closed piecewise smooth curve \(C\) to the value of a(n) _____ integral over the simply connected closed region \(R\) enclosed by \(C\).

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If the assumptions of Green’s Theorem are met, then \(\oint_{C}(P(x,y) \,dx+Q(x,y) \,dy)=\iint\limits_{R}\) ( _____) \(dx\,dy\).

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True or False The symbol \(\oint_C\) indicates that a simple closed curve \(C\) is to be traversed in a clockwise direction.

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If \(C\) is a piecewise-smooth, simple closed curve, then \( \dfrac{1}{2}\oint_{C}(-y\,dx+x\,dy)\) equals the _____ of the closed simply connected region \(R\) enclosed by \(C\).

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\(\oint_{C}y\,dx\)

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\(\oint_{C}x\,dy\)

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\(\oint_{C}[xy\,dx+(x+y)\,dy]\)

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\(\oint_{C}(3y\,dx-2x\,dy)\)

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\(\oint_{C}(xy^{2}\,dx+x^{2}y\,dy)\)

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\(\oint_{C}\left[ y\sin (xy) \,dx+x\sin (xy) \,dy\right] \)

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\(\oint_{C}(-x^{2}y\,dx+y^{2}x\,dy)\)

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\(\oint_{C}[y(x^{2}+y^{2})\,dx-x(x^{2}+y^{2})\,dy]\)

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\(\oint_{C}[(x^{2}-y^{3})\,dx+(x^{2}+y^{2})\,dy]\)

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\(\oint_{C}[(xy^{3}+\sin x)\,dx+(x^{2}y^{2}+4x)\,dy]\)

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\(\oint_{C}[(x^{2}+y)\,dx+(x-y^{2})\,dy]\), where \(C\) is the boundary of the region \(R\) bounded by \(y=x^{3/2}\), the \(x\)-axis, and \(x=1\)

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\(\oint_{C}[(4x^{2}-8y^{2})\,dx+(y-6xy)\,dy]\), where \(C\) is the boundary of the region \(R\) bounded by \(y=\sqrt{x}\) and \(y=x^{2}\)

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\(\oint_{C}[(x^{3}-x^{2}y)\,dx+xy^{2}\,dy]\), where \(C\) is the boundary of the region \(R\) bounded by \(y=x^{2}\) and \(x=y^{2}\)

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\(\oint_{C}[(x^{2}-y^{2})\,dx+xy\,dy]\), where \(C\) is the boundary of the region \(R\) bounded by \(x=y^{2}\) and \(x=1\)

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\(\oint_{C}\left[ \dfrac{1}{y}\,dx+\dfrac{1}{x}\,dy\right] \), where \(C\) is the boundary of the region \(R\) bounded by \(y=1\), \(x=9\), and \(y= \sqrt{x}\)

1015

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\(\oint_{C}(x^{2}y\,dx-y^{2}x\,dy)\), where \(C\) is the boundary of the region \(R\) bounded by \(y=\sqrt{a^{2}-x^{2}}\) and \(y=0\)

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\(C\): \(x^{2}+y^{2}=4\)

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\(C\): \((x-2)^{2}+(y+3)^{2}=1\)

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\(C\): \(x^{2}+4y^{2}=4\)

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\(C\): \(3x^{2}+6y^{2}=4\)

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Bounded by \(y=x^{2}\) and \(y=x+2\)

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Bounded by \(y=x^{2}-1\) and \(y=0\)

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Bounded by \(y=x^{3}\) and \(y=x^{2}\)

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Bounded by \(y=2\sqrt{x}\) and \(y=\dfrac{x^{2}}{4}\)

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Bounded by \(y=\dfrac{1}{\sqrt{x-1}}\), \(y=0\), \(x=2\), and \(x=5\)

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Bounded by \(y=x^{3/2}\) and \(y=x\)

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Bounded by the line \(x+y=3\) and the hyperbola \(xy=2\)

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Bounded by the hyperbola \(xy=\sqrt{3}\) and the circle \(x^{2}+y^{2}=4\), in the first quadrant only

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Area Find the area under one arch of the cycloid \(\mathbf{r} (t)=\left[ 2\pi t-\sin (2\pi t) \right] \,\mathbf{i}+\left[ 1-\cos (2\pi t) \right] \,\mathbf{j},\) \(0\leq t\leq 1\).

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Area Find the area enclosed by the hypocycloid \(x^{2/3}+y^{2/3}=1\).(Hint: Use the parametric equations \(x=\cos ^{3}t \), \(y=\sin ^{3}t\).)

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A folium of Descartes is given by the parametric equations \(x(t) =\dfrac{6t}{1+t^{3}},\) \(y(t) =\dfrac{ 6t^{2}}{1+t^{3}}.\) Use Green’s Theorem to show that the area \(A\) enclosed by the loop of this folium of Descartes is \(6\).

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The region bounded by the \(x\)-axis and the semi-circle \(y=\sqrt{4-x^{2}}\)

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The region bounded by the parabola \(y=x^{2}\) and the line \(y=1\)

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The region bounded by the triangle with vertices at \((0,0) ,\) \((1,0) ,\) and \(( 0,1)\)

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The region bounded by the trapezoid with vertices at \((0,0)\), \((1,0) ,\) \(( 1,2) ,\) and \((0,1)\)

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The region bounded by the coordinate axes and the hypocycloid \(x=a\cos ^{3}t\), \(y=a\sin ^{3}t\), \(0\leq t\leq \dfrac{\pi }{2}\)

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Work  Use Green’s Theorem to find the work done by the variable force \(\mathbf{F}(x,y) =x\mathbf{i}+y^{2}\mathbf{j}\) in moving an object along the curve \(x(t) =t\), \(y(t) =t^{2},\) \(0\leq t\leq 4,\) and then returning to the point \((0,0) \) along the line \(x(t) =t\), \(y(t) =4t\).

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Work  Use Green’s Theorem to find the work done by the variable force \(\mathbf{F}(x,y) =\left( e^{x}+2y\right) \mathbf{i}+\left( e^{y}-x\right) \mathbf{j}\) in moving an object along the closed path \(x^{2}+y^{2}=9\).

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Find \[ \oint_{\!\!\!C} \left\{ [\ln(1+x^2) -x^2\cos y]\,dx + \left(\dfrac{x^3}{3}\cos y +e^{\cos y} \right)dy \right\} , \] where \(C\) is the boundary of the region enclosed by \(y=x^3\), \(x=1\), and the \(x\)-axis.

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Find \(\oint_{C} [y^3\,dx-x^3\,dy] \), where \(C\) is the boundary of the multiply-connected region \(1 \leq x^2 + y^2 \leq 9\) and \(C\) is traversed in the counter-clockwise direction.

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\(\oint_{C}\big(xe^{x^{2}+y^{2}}\,dx+ye^{x^{2}+y^{2}}\,dy\big)\), where \(C\) is any piecewise-smooth, simple closed curve in the plane

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\(\oint_{C}[e^{xy}(xy+1)\,dx+x^{2}e^{xy}\,dy]\), where \(C\) is any piecewise-smooth, simple closed curve in the plane

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\(\oint_{C}(e^{x}\sin y\,dx+e^{x}\cos y\,dy)\), where \(C\) is any piecewise-smooth, simple closed curve in the plane

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\(\oint_{C}\left[ \dfrac{x\,dx}{(x^{2}+y^{2})^{1/2}}+\dfrac{ y\,dy}{(x^{2}+y^{2})^{1/2}}\right] \), where \(C\) is any piecewise-smooth, simple closed curve not containing the origin in its interior

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Green’s Theorem  Complete the proof of Green’s Theorem for \(x\) -simple and \(y\)-simple regions by showing \(\oint_{C}Q(x,y)\,dy=\iint \limits_{ R}\dfrac{\partial Q}{\partial x}\,dx\,dy\).

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Use Green’s Theorem for finding the area \(A\) of the region \(R\) enclosed by simple closed piecewise smooth curve \(C\) to show that \(A=\oint_{C}x\,dy\) and \(A=\oint_{C}(-y)\,dx\).

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Consider the integral \(\int_{C}\dfrac{y\,dx-x\,dy}{ (x^{2}+y^{2})^{3/2}}\). For which simple closed, piecewise-smooth curves \(C\) can Green’s Theorem be applied to transform this line integral to a double integral?

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Center of Mass A homogeneous lamina occupies a region in the \( xy\)-plane with boundary \(C\) (oriented counterclockwise). Show that its center of mass has the coordinates given below, where \(A\) is the area of the region. \[ \bar{x}=\dfrac{1}{2A}\oint\nolimits_{\!\!C}x^{2}\,dy\qquad \bar{y}=\dfrac{-1}{2A} \oint\nolimits_{\!\!C}y^{2}\,dx \]

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Moment of Inertia Use Problem 52 to show that the moment of inertia about the origin is \[ I_{o}=\dfrac{1}{3}\int_{C}(x^{3}\,dy-y^{3}\,dx) \] (Assume that the mass density of the lamina is \(1\).)

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Let \(R\) be a simply connected closed region in the \(xy\) -plane, and let \(R′ \) be a simply connected closed region in the \(st\) -plane. Suppose \(x=x(s,t)\) and \(y=y(s,t)\) are differentiable and that, as \((s,t)\) is allowed to vary throughout \(R′\), \((x,y)\) varies throughout \(R \) in a one-to-one correspondence. Suppose further that \(R\) has boundary \(C\) , which is a piecewise-smooth closed curve, and \(R′ \) has boundary \(C′ \), which is a piecewise-smooth closed curve. Finally, suppose that as \((s,t)\) moves along \(C′ \), \((x(s,t),y(s,t))\) moves along \(C\). Show that \[ \iint\limits_{R}\,dx\,dy=\iint\limits_{R^{\prime }} (x_{s}y_{t}-x_{t}y_{s}) \,ds\,dt \]

[Hint: Using \(\iint\limits_{R}\,dx\,dy=\dfrac{1}{2}\oint_{C}(x\,dy-y\,dx)\), write the integral on the right in terms of \(s\) and \(t\). Then apply Green’s Theorem to the resulting integral, which will have the form \(\oint_{C’ }(P\,ds+Q\,dt)\).]

1016

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Transformation of a Double Integral Let \(\phi (x,y)\) be continuous and let \(P(x,y)=-\int \phi (x,y)\,dy\) and \(Q(x,y)=\int \phi (x,y)\,dx\). Generalize the result of Problem 54 using \(P\) and \(Q\) to show that \[ \iint\limits_{R}\phi (x,y)\,dx\,dy=\iint\limits_{R^{\prime }}\phi (x(s,t),y(s,t))(x_{s}y_{t}-x_{t}y_{s}) \,ds\,dt \] Go through a similar process as in Problem 54, using Green’s Theorem on the integral \(\oint_{C}P\,dx+Q\,dy\).