Chapter 14: Q 29. (page 1107)

Evaluate each of the vector field line integrals over the indicated curves.
$F (x, y) = i - j$ with $C$ the curve with equation $y - x^{2} = 1$ for $5 \leq x \leq 10$ .

Short Answer

Expert verified

The required line integral is $- 70$ .

Step by step solution

Given Information

It is given that $F (x, y) = i - j$

The curve is parametrized by $y - x^{2} = 1$ for $5 \leq x \leq 10$

Solving the parametric equations

The given curve is parametrize by

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

$\Rightarrow x^{'} (t) = 1, y^{'} (t) = 2 t$

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

Also

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

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

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

Solving the line integral

The formula for line integral is

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

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

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

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

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

$= - 90 + 20$

$= - 70$

The line integral is $- 70$ .

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Evaluate each of the vector field line integrals over the indicated curves.
$F (x, y) = i - j$ with $C$ the curve with equation $y - x^{2} = 1$ for $5 \leq x \leq 10$ .

Short Answer

Step by step solution

Given Information

Solving the parametric equations

Solving the 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)=i-j with C the curve with equationy-x2=1for5≤x≤10.

Short Answer

Step by step solution

Given Information

Solving the parametric equations

Solving the line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Probability and Statistics

Mechanics Maths

Applied Mathematics

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

Evaluate each of the vector field line integrals over the indicated curves.
$F (x, y) = i - j$ with $C$ the curve with equation $y - x^{2} = 1$ for $5 \leq x \leq 10$ .