Q. 23 In Exercises 21鈥�28, evaluate t... [FREE SOLUTION]

Chapter 14: Q. 23 (page 1106)

In Exercises 21鈥�28, evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y) = x^{2} + y^{2}$ , with C the unit circle traversed counterclockwise.

Short Answer

Expert verified

The multivariate line integral of the given function over the specified curve is ${鈭�}_{C} f (x, y) d s = 2 蟺$ .

Step by step solution

Step 1. Given Information

In the given exercises we have to evaluate the multivariate line integral of the given function over the specified curve.

$f (x, y) = x^{2} + y^{2}$ , with C the unit circle traversed counterclockwise.

Step 2. The given function is f(x,y)=x2+y2

So the line integral is

${鈭�}_{C} f (x, y) d s = {鈭�}_{a}^{b} f (r (t)) \sqrt{(x' {(t))}^{2} + (y' {(t))}^{2}} d t$

Parametrization: $x = \cos (t), y = \sin (t), 0 鈮� t 鈮� 2 蟺 . So, d x = - \sin (t) d t, d y = \cos t$ .

Now finding $f (r (t)$

$f (r (t)) = x^{2} + y^{2} f (r (t)) = \cos t^{2} + \sin t^{2} f (r (t)) = 1$

Step 3. Now solving the ∫Cf(x,y)ds=∫abf(r(t))(x'(t))2+(y'(t))2dt

${鈭�}_{C} f (x, y) d s = {鈭�}_{0}^{2 蟺} 1 \sqrt{{(\cos t)}^{2} + {(\sin t)}^{2}} d t {鈭�}_{C} f (x, y) d s = {鈭�}_{0}^{2 蟺} 1 \sqrt{\cos^{2} t + \sin^{2} t} d t {鈭�}_{C} f (x, y) d s = {鈭�}_{0}^{2 蟺} 1 \sqrt{1} d t {鈭�}_{C} f (x, y) d s = {鈭�}_{0}^{2 蟺} d t {鈭�}_{C} f (x, y) d s = {[t]}_{0}^{2 蟺} {鈭�}_{C} f (x, y) d s = [2 蟺 - 0] {鈭�}_{C} f (x, y) d s = 2 蟺$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Exercises 21鈥�28, evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y) = x^{2} + y^{2}$ , with C the unit circle traversed counterclockwise.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The given function is f(x,y)=x2+y2

Step 3. Now solving the ∫Cf(x,y)ds=∫abf(r(t))(x'(t))2+(y'(t))2dt

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 21鈥�28, evaluate the multivariate line integral of the given function over the specified curve.f(x,y)=x2+y2, with C the unit circle traversed counterclockwise.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. The given function is f(x,y)=x2+y2

Step 3. Now solving the ∫Cf(x,y)ds=∫abf(r(t))(x'(t))2+(y'(t))2dt

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

In Exercises 21鈥�28, evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y) = x^{2} + y^{2}$ , with C the unit circle traversed counterclockwise.