Verify the circumference of the circle formula for a circle of radius r using the arc length formula.How do I complete this problem? Do I use the ellipse equation or the proper circle equation with (x, h) and (y,k)?

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Rebekah Zimmerman · Accepted Answer

Do it with half a circle that we can assume is centered at the origin:      x    2    +      y    2    =      r    2      ⟹    y  =            r      2        −          x      2          ,          y    ′    =  −      x                  r        2            −              x        2            Thus (the function&#039;s even)  I  :=      ∫          −      r        r        1    +                  x        2                              r          2                −                  x          2                        d  x  =  2      ∫    0    r              d      x              1      −                        (                      x            r                    )                2            Substitution  x  =  r  sin  ⁡  u    ⟹    d  x  =  r  cos  ⁡  u    ⟹    I  =  2      ∫    0          π              /            2                  r      cos      ⁡      u            d      u              cos      ⁡      u        =  π  rUsing parametrization in two variables: we can write the same circle above as  (  x  (  t  )    ,    y  (  t  )  )    ,      with      x  (  t  )  =  r  cos  ⁡  t    ,      y  =  r  sin  ⁡  t    ,      0  ≤  t  ≤  2  πand thus the arclength is given by      ∫    0          2      π                  x      ′        (    t          )      2        +          y      ′        (    t          )      2        d  t  =      ∫    0          2      π                  r      2              sin      2        ⁡    t    +          r      2              cos      2        ⁡    t    d  t  =  r      ∫    0          2      π        d  t  =  2  π  r

Verify the circumference of the circle formula for a circle

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