Chapter 14: Q 14. (page 1106)

Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of $t$ , where $F (x, y, z) = {z y - x, x z - y, x y - z ⟩$ and $r (t) = ⟨ t, \cos t, \sin t ⟩$ for $a \leq t \leq b$

Short Answer

Expert verified

The required answer is $\int_{a}^{b} (\frac{1}{2} \sin 2 t - t + t \cos 2 t) d t$ .

Step by step solution

Given Information

It is given that integral is $F (x, y, z) = ⟨ z y - x, x z - y, x y - z ⟩$

$r (t) = ⟨ t, \cos t, \sin t ⟩$ for $a \leq t \leq b$

Determination of F(x(t),y(t),z(t))

The line Integral formula is $\int_{C} F \cdot d r = \int_{C} F (x (t), y (t), z (t)) \cdot r^{'} (t) d t$

Parametric Equations are $x (t) = t, y (t) = \cos t, z (t) = \sin t$ for $a \leq t \leq b$

Hence, $r^{'} (t) = <x^{'} (t), y^{'} (t), z^{'} (t)>$

$= ⟨ 1, - \sin t, \cos t ⟩$

$\Rightarrow F (x (t), y (t), z (t)) = ⟨ \sin t \cos t - t, t \sin t - \cos t, t \cos t - \sin t ⟩$

Expression of Line Integral

The line integral is

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

$= \int_{a}^{b} ⟨ \sin t \cos t - t, t \sin t - \cos t, t \cos t - \sin t ⟩ \cdot ⟨ 1, - \sin t, \cos t ⟩ d t$

$= \int_{a}^{b} [(\sin t \cos t - t) (1) + (t \sin t - \cos t) (- \sin t) + (t \cos t - \sin t) (\cos t)] d t$

$= \int_{a}^{b} (\sin t \cos t - t - t \sin^{2} t + \sin t \cos t + t \cos^{2} t - \sin t \cos t) d t$

$= \int_{a}^{b} (\sin t \cos t - t + t (\cos^{2} t - \sin^{2} t)) d t$

$= \int_{a}^{b} (\frac{1}{2} \sin 2 t - t + t \cos 2 t) d t$

Hence, required line integral is $= \int_{a}^{b} (\frac{1}{2} \sin 2 t - t + t \cos 2 t) d t$

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Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of $t$ , where $F (x, y, z) = {z y - x, x z - y, x y - z ⟩$ and $r (t) = ⟨ t, \cos t, \sin t ⟩$ for $a \leq t \leq b$

Short Answer

Step by step solution

Given Information

Determination of F(x(t),y(t),z(t))

Expression of Line Integral

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Write∫CF(x,y,z)·drexplicitly as an integral oft, whereF(x,y,z)={zy-x,xz-y,xy-z⟩andr(t)=⟨t,cost,sint⟩fora≤t≤b

Short Answer

Step by step solution

Given Information

Determination of F(x(t),y(t),z(t))

Expression of Line Integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Write $\int_{C} F (x, y, z) \cdot d r$ explicitly as an integral of $t$ , where $F (x, y, z) = {z y - x, x z - y, x y - z ⟩$ and $r (t) = ⟨ t, \cos t, \sin t ⟩$ for $a \leq t \leq b$