## DBsubject('Calculus') ## DBchapter('Fundamental Theorems of Vector Analysis') ## DBsection('Greens Theorem') ## KEYWORDS('calculus') ## TitleText1('Calculus: Early Transcendentals') ## EditionText1('2') ## AuthorText1('Rogawski') ## Section1('17.1') ## Problem1('9') ## Author('JustAsk - Kobi Fonarov') ## Institution('W.H.Freeman') ## UsesAuxiliaryFiles('image_17_1_9.png') DOCUMENT(); loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); loadMacros("PGauxiliaryFunctions.pl"); loadMacros("PGgraphmacros.pl"); loadMacros("PGchoicemacros.pl"); TEXT(beginproblem()); $a=random(2,9);$b=random(2,9); $b2=$b**2; $fxdeg=random(2,9);$GreenResult=Real($a*$b2); $answer=-Real($a*$b2);$boundary="\partial\mathcal{D}"; $curve="\mathcal{C}";$domain="\mathcal{D}"; $FF="\mathbf{F}"; TEXT('') if$displayMode eq 'HTML_jsMath'; Context()->texStrings; BEGIN_TEXT \{ textbook_ref_exact("Rogawski ET 2e", "17.1","9") \} $PAR Use Green's Theorem to evaluate the line integral of $$FF = \left< x^{fxdeg}, a x\right>$$$PAR around the boundary of the parallelogram in the following figure (note the orientation). $PAR$BR \{image("image_17_1_9.png", width=>234, height=>157)\} With $$x_0=b$$ and $$y_0=b$$. $PAR $$\int_{curve} x^{fxdeg} \,dx+a x \,dy =$$ \{ans_rule()\}$PAR END_TEXT ANS($answer->cmp); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION'));$PAR \$SOL First note that the orientation of the boundary curve is clockwise. We will use Green's Theorem remembering that the boundary curve must be oriented counterclockwise. We have $$P= x^{fxdeg}$$ and $$Q=a x$$, therefore $\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=a-0=a$ Hence, Green's Theorem implies $\int_{boundary} x^{fxdeg} \,dx+a x \,dy = \iint_{domain} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \,dA =$$\iint_{domain} a \,dA=a \iint_{domain} \,dA = a\, \mathrm{Area}(domain)=a \cdot b2=GreenResult$ So now accounting for the orientation, $\[\int_{curve} x^{fxdeg} \,dx+a x \,dy = -\[\int_{boundary} x^{fxdeg} \,dx+a x \,dy =answer$ END_SOLUTION ENDDOCUMENT();