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Chapter 14: Q 14. (page 1154)

Evaluate the line integral of the given function over the specified curve.
${鈭�}_{c} F (x, y, z) 路 d r$ where $F (x, y, z) = y z i + x z j + x y k$ and C is the curve parametrized by $x = t^{2}, y = t^{- 2}, z = t$ for $1 鈮� t 鈮� 3$ .

Short Answer

Expert verified

${鈭�}_{c} F (x, y, z) . d r = 2$

Step by step solution

01

Given Information

The given expression is $F (x, y, z) = y z i + x z j + x y k$ and $r (t) = (t^{2}, t^{- 2}, t)$ and

$1 鈮� t 鈮� 3$ .

02

Simplification

The curve is parametrized as

$r (t) = (t^{2}, t^{- 2}, t)$

Taking derivative

$r^{'} (t) = (2 t, - 2 t^{- 3}, 1)$

Putting vales of $x, y, z$

role="math" localid="1652331665945" $F (x (t), y (t), z (t)) = t^{- 2} (t) i + t^{2} (t) j + t^{2} t^{- 2} k$

$= t^{- 1} i + t^{3} j + k$

$= (t^{- 1}, t^{3}, 1)$

The line integral is evaluated as
${鈭�}_{C} F (x, y, z) . d r = {鈭�}_{C} F (x (t), y (t), z (t)) 路 r^{'} (t) d t$

$= {鈭�}_{1}^{3} (t^{- 1}, t^{3}, 1) 路 (2 t, - 2 t^{- 3} . 1) d t$

$= {鈭�}_{1}^{3} [t^{- 1} (2 t) + t^{3} (- 2 t^{- 3}) + 1 (1)] d t$

$= {鈭�}_{1}^{3} (2 - 2 + 1) d t$

$= {[t]}_{1}^{3}$

$= 3 - 1$

$= 2$

${鈭�}_{c} F (x, y, z) . d r = 2$

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