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\noindent {\bf Problem statement} Suppose that $C_1$ and $C_2$ are two
curves in the plane, given parametrically by
 
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\centerline{$C_1 \colon\ \left\{\eqalign{x(t)&=0\cr y(t)&=2t-1\cr}\right. {\rm\
for\ } 0\le t \le 1\,;\quad
C_2 \colon\ \left\{\eqalign{x(t)&=-\cos t\cr y(t)&=\sin t\cr}\right. {\rm\
for\ } -{\pi\over 2}\le t \le {\pi\over 2}\,.
$}

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\noindent a) Sketch these curves.
 
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\noindent b) If $f$ is a function defined in the plane, let $I_1(f)$
and $I_2(f)$ be the line integrals of $f$ over these curves with
respect to arc length: $\displaystyle I_1(f)=\int_{C_1}f(x,y)\,ds$ and
$\displaystyle I_2(f)=\int_{C_2}f(x,y)\,ds$.  In each case below
determine which of $I_1(f)$ and $I_2(f)$ is greater or whether they
are equal (that is, whether $I_1(f)>I_2(f)$, $I_1(f)<I_2(f)$ or
$I_1(f)=I_2(f)$) {\it without evaluating the integrals}.  Explain your
reasoning carefully.  Then check your answer by computing the
integrals.
 
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\centerline{ i) $f(x,y)=17$;\qquad\qquad ii) $f(x,y)=x$;
   \qquad\qquad iii) $f(x,y)=y$.}










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