Chapter 14: Q 14. (page 1154)

Evaluate the line integral of the given function over the specified curve.
$\int_{c} F (x, y, z) \cdot d r$ 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 $1 \leq t \leq 3$ .

Short Answer

Expert verified

$\int_{c} F (x, y, z) . d r = 2$

Step by step solution

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

$1 \leq t \leq 3$ .

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
$\int_{C} F (x, y, z) . d r = \int_{C} F (x (t), y (t), z (t)) \cdot r^{'} (t) d t$

$= \int_{1}^{3} (t^{- 1}, t^{3}, 1) \cdot (2 t, - 2 t^{- 3} . 1) d t$

$= \int_{1}^{3} [t^{- 1} (2 t) + t^{3} (- 2 t^{- 3}) + 1 (1)] d t$

$= \int_{1}^{3} (2 - 2 + 1) d t$

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

$= 3 - 1$

$= 2$

$\int_{c} F (x, y, z) . d r = 2$

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Evaluate the line integral of the given function over the specified curve.
$\int_{c} F (x, y, z) \cdot d r$ 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 $1 \leq t \leq 3$ .

Short Answer

Step by step solution

Given Information

Simplification

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Evaluate the line integral of the given function over the specified curve.∫cFx,y,z·drwhere Fx,y,z=yzi+xzj+xykand C is the curve parametrized byx=t2,y=t-2,z=tfor1≤t≤3.

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

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Pure Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Evaluate the line integral of the given function over the specified curve.
$\int_{c} F (x, y, z) \cdot d r$ 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 $1 \leq t \leq 3$ .