Chapter 14: Q 31. (page 1107)

Evaluate each of the vector field line integrals over the indicated curves.
$F (x, y)$ and $C$ are as in Exercise 29, but $C$ is traversed in the reverse direction, from $x = 10$ to $x = 5$

Short Answer

Expert verified

The required integral is $\int_{C} F \cdot d r = 70$ .

Step by step solution

Given Information

The given vector field is $F (x, y) = i - j$

The curve is parametrized as

$y - x^{2} = 1$ for $10 \leq x \leq 5$

Solving parametrized equations

The parametric equations of curve is:

$x (t) = t, y (t) = 1 + t^{2}, 10 \leq t \leq 5$

role="math" localid="1653593890857" $\Rightarrow x^{'} (t) = 1, y^{'} (t) = 2 t$

Hence,

$C : r (t) = <t, 1 + t^{2}>$

Given is $F (x (t), y (t)) = i - j$

and

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

$= ⟨ 1, 2 t ⟩$

$= (1) i + (2 t) j$

Finding line integral

The formula is

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

$= \int_{10}^{5} [(1) i + (- 1) j] \cdot [(1) i + (2 t) j] d t$

$= \int_{10}^{5} (1 - 2 t) d t$

$= [t - {t^{2}|}_{10}^{5}]$

$= [5 - 25] - [10 - 100]$

$= - 20 + 90$

$= 70$

Therefore, $\int_{C} F \cdot d r = 70$

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Evaluate each of the vector field line integrals over the indicated curves.
$F (x, y)$ and $C$ are as in Exercise 29, but $C$ is traversed in the reverse direction, from $x = 10$ to $x = 5$

Short Answer

Step by step solution

Given Information

Solving parametrized equations

Finding line integral

One App. One Place for Learning.

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Evaluate each of the vector field line integrals over the indicated curves.F(x,y)and Care as in Exercise 29, but Cis traversed in the reverse direction, fromx=10tox=5

Short Answer

Step by step solution

Given Information

Solving parametrized equations

Finding line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Geometry

Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate each of the vector field line integrals over the indicated curves.
$F (x, y)$ and $C$ are as in Exercise 29, but $C$ is traversed in the reverse direction, from $x = 10$ to $x = 5$