Find arc length given x = t sin t , y = t cos t and 0 ≤ t ≤ 1 ?

Question

Find arc length given       x    =    t          sin      t      ,       y    =    t          cos      t       and       0    ≤    t    ≤    1  ?

Gerald Dickerson · Accepted Answer

x    =    t          sin      t            x    ′    =          sin      t        +    t          cos      t            y    =    t          cos      t            y    ′    =          cos      t        -    t          sin      t      Arc length is given by:      L    =                  ∫        0        1                                                  (                          sin              t                        +            t                          cos              t                        )                    2                +                              (                          cos              t                        -            t                          sin              t                        )                    2                            d      t      Expand and simplify:      L    =                  ∫        0        1                            1        +                  t          2                            d      t      Apply the substitution       t    =          tan      θ      :      L    =                  ∫        0                                            tan                              -                1                                                    (              1              )                                                          sec        3            θ        d    θ  This is a recognized integral. If you don&#039;t remember it, look it up in a table of integrals or use integration by parts:      L    =          1      2                                [                      sec            θ                                tan            θ                    +          ln                      |                          sec              θ                        +                          tan              θ                        |                    ]                0                                            tan                              -                1                                                    (              1              )                                          Insert the limits of integration:      L    =          1      2              (              2            +              ln                  (          1          +                      2                    )                    )

What is the arc length of the curve given by r(t)=(1, t, t^2) on t in [0, 1]?

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