In training for a swim meet, Logan swam 750 meters in 1/3 of an hour. His swimming partner Mila, swam 2/3 of Logan's distance in
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The length of a curve <em>C</em> parameterized by a vector function <em>r</em><em>(t)</em> = <em>x(t)</em> i + <em>y(t)</em> j over an interval <em>a</em> ≤ <em>t</em> ≤ <em>b</em> is
In this case, we have
<em>x(t)</em> = exp(<em>t</em> ) + exp(-<em>t</em> ) ==> d<em>x</em>/d<em>t</em> = exp(<em>t</em> ) - exp(-<em>t</em> )
<em>y(t)</em> = 5 - 2<em>t</em> ==> d<em>y</em>/d<em>t</em> = -2
and [<em>a</em>, <em>b</em>] = [0, 2]. The length of the curve is then
Recall that
There are three cases to consider:
(1) When
, we have 
and 
, so
(2) When
and 
, we get 
and , so
(3) When
, we have and 
, so
So
There are three cases to consider:
(1) When
(2) When
(3) When
So