How to find the length of the polar curve r = cos 3 ( θ 3 ) ?

Question

How to find the length of the polar curve       r    =                  cos        3                    (                  θ          3                )            ?

Hunter Hendricks · Accepted Answer

We using the chain rule.By Chain Rule,                    d        r                    d        θ              =    3                  cos        2                    (                  θ          3                )              ⋅          [      -              sin                  (                      θ            3                    )                    ]        ⋅          1      3      by cleaning up a bit,      =    -                  cos        2                    (                  θ          3                )                    sin              (                  θ          3                )            Note that       θ   goes from 0 to       3    π   to complete the loop once.Let us now calculate the curve&#039;s length L.      L    =                  ∫        0                  3          π                                              r          2                +                              (                                          d                r                                            d                θ                                      )                    2                      d    θ        =                  ∫        0                  3          π                                                          cos            6                                (                          θ              3                        )                          +                              cos            4                                (                          θ              3                        )                                                sin            2                                (                          θ              3                        )                                d    θ  by pulling                     cos        2                    (                  θ          3                )             out of the square-root,      =                  ∫        0                  3          π                                    cos        2                    (                  θ          3                )                                                  cos            2                                (                          θ              3                        )                          +                              sin            2                                (                          θ              3                        )                                d    θ  by                     cos        2            θ        =          1      2              (      1      +              cos        2            θ      )       and                     cos        2            θ        +                  sin        2            θ        =    1  ,      =          1      2                      ∫        0                  3          π                            [      1      +              cos                  (                                    2              θ                        3                    )                    ]        d    θ        =          1      2                                [          θ          +                      3            2                                sin                          (                                                2                  θ                                3                            )                                ]                0                  3          π                          =          1      2              [      3      π      +      0      -              (        0        +        0        )            ]        =                  3        π            2

How to find the length of the polar curve r = cos 3 ( θ...

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