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    \bar{r}(t)=(-ln(cos(t)), t), with t\in [0,\frac{\pi}{4}], is a parameterised curve in the plane. Calculate its length. Find the arclength parameter s as a function of t.
    So I got \displaystyle \int_0^\frac{\pi}{4} ||\frac{d\bar{r}}{dt}||dt is the length.

    \frac{d\bar{r}}{dt}=(tan(t),1)

    ||\frac{d\bar{r}}{dt}||=\sqrt{ta  n^{2}(t)+1}

    Therefore \displaystyle L= \int_0^\frac{\pi}{4}\sqrt{tan^{2  }(t)+1} dt

    Is this even integrable? Have I found the length or the arclength? The variation in definition between the two in my notes seems hazy.
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    (Original post by RamocitoMorales)
    So I got \displaystyle \int_0^\frac{\pi}{4} ||\frac{d\bar{r}}{dt}||dt is the length.

    \frac{d\bar{r}}{dt}=(tan(t),1)

    ||\frac{d\bar{r}}{dt}||=\sqrt{ta  n^{2}(t)+1}

    Therefore \displaystyle L= \int_0^\frac{\pi}{4}\sqrt{tan^{2  }(t)+1} dt

    Is this even integrable? Have I found the length or the arclength? The variation in definition between the two in my notes seems hazy.
    Sec^2 (x) = 1 + tan^2 (x)
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    (Original post by joostan)
    Sec^2 (x) = 1 + tan^2 (x)
    Ah, thanks. I calculated L=\ln{(\sqrt{2}+1)}. Have I answered the question? Are the 'length' and 'arc length' synonymous? Should I change my 'L' for an 's(t)'?
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    Anyone? :dontknow:
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    (Original post by RamocitoMorales)
    Ah, thanks. I calculated L=\ln{(\sqrt{2}+1)}. Have I answered the question? Are the 'length' and 'arc length' synonymous? Should I change my 'L' for an 's(t)'?
    The length of the curve and arc length are synonymous.

    For the curve from t=0 to t=pi/2 you will have L=

    Not sure why it asks "Find the arclength parameter s as a function of t" at the end, unless that's suggesting how to do it.
 
 
 
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