Evaluate the line integral, where C is the given curve. ∫ C x y z 2 d s. C is the line segment from (-1,2, 0) to(1, 3,3).

Question

Evaluate the line integral, where C is the given curve.       ∫    C    x  y      z    2    d  s. C is the line segment from (-1,2, 0) to(1, 3,3).

Alma Garner · Accepted Answer

The line segment can be represented with parametric equations:  x  =  −  1  +  2  t  ,  y  =  2  +  t  ,  z  =  3  t, for   0  ≤  t  ≤  1Evaluating the given line integral yields:      ∮    C    f  (  x  ,  y  ,  z  )  d  s  =      ∫    a    b    f  (  x  (  t  )  ,  y  (  t  )  ,  z  (  t  )  )      (                  d        x                    d        t                    )      2        +    (                  d        y                    d        t                    )      2        +    (                  d        z                    d        t                    )      2        d  t        ∮    C    x  y      z    2    d  s  =      ∫    0    1    (  −  1  +  2  t  )  (  2  +  t  )  (  3  t      )    2        (    2          )      2        +    (    1          )      2        +    (    3          )      2        d  t    =      ∫    0    1    (  18      t    4    +  27      t    3    −  18      t    2    )      14    d  t    =      14        ∫    0    1    (  18      t    4    +  27      t    3    −  18      t    2    )  d  t    =      14    (      18    5        t    5    +      27    4        t    4    −  6      t    3    )      ]    0    1    =            87              14              20

Evaluate the line integral, where C is the given curve. int_C xyz^2 ds. C is the line segment from (-1,2, 0) to(1, 3,3).

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