**Problem Statement:**Use Green's Theorem to evaluate the line integral along the given positively oriented curve.\[ \oint_{C} xe^{-5x} \, dx + \left( x^4 + 2x^2y^2 \right) \, dy \]**Note:**- $ C $ is the boundary of the region between the circles $ x^2 + y^2 = 9 $ and $ x^2 + y^2 = 25 $.### Explanation:**Green's Theorem** connects a line integral around a simple closed curve $ C $ to a double integral over the plane region $ D $ bounded by $ C $:\[\oint_{C} \left( M \, dx + N \, dy \right) = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA\]Here:- $ M(x, y) = xe^{-5x} $- $ N(x, y) = x^4 + 2x^2y^2 $**Region $ C $**: The region is an annular area (the area between two circles) with radii determined by:- The inner circle: $ x^2 + y^2 = 9 $, with radius 3.- The outer circle: $ x^2 + y^2 = 25 $, with radius 5.**Graphs/Diagrams:**- Assume two concentric circles are drawn on a coordinate plane.- The inner circle has a radius of 3 and the outer circle has a radius of 5.- The region $ C $ is the shaded area between these two circles.To solve, apply Green's Theorem to convert the line integral into a double integral over the annular region. Calculate $ \frac{\partial N}{\partial x} $ and $ \frac{\partial M}{\partial y} $, and integrate over the given region to find the area.

Answered: Image /qna-images/answer/ddde3b41-2ffa-4b95-9c45-15df7aabd850.jpg

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. -5x хе (x* + 2x3y²) dy dx + Cis the boundary of the region between the circles x2 + y? = 9 and x2 + y2 = 25 %3D

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. -5x хе (x* + 2x3y²) dy dx + Cis the boundary of the region between the circles x2 + y? = 9 and x2 + y2 = 25 %3D

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Evaluating Polar In In Exercises 1-16, change s C- polar integral. Then evaluat ep VI- 1. dy dx

Q: Evaluate the line integral (2 yi + (r- 2)j+ zyzk) · dr where C is the parametric curve r(t) = ti +…

Q: 3) Evaluate the iterated integral by converting the to polar coordinates -√₂x-x² a) √²ty² dydx…

A: Given that, The integrated integral: (a) ∫02∫02x-x2x2+y2dydx, such that , the region is…

Q: Evaluate | f(x,y) ds where Cis the portion of the curve y = In x from the point (1,0) to the point…

Q: Q2: I| (x? +y?) dxdy a) Use the polar coordinate to find the integral 0 0 1/16 1/2 b) By using the…

A: In the given question we have to find the integral value of the function.

Q: 6. Tangent Plane dro find an equatin the Plane and a set Line equatin g tho of Tangent * 9 o the…

A: We have to find the equation of tangent plane and set of equation of the normal line to the given…

Q: X- An ant is moving on the surface z = x² + xy + y² such that its - and y-coordinates satisfy cost,…

Q: QI/ Find the value of the line integral , F.dr when F-yi-xyj and C is the circular are in figure…

Q: Plot the slope field of y=x+y using slopes m = -3, -2,-1,0, 1, 2. Sketch integral curves for…

A: First we plot the slope field and then plot the curve .

Q: How do I evaluate this?

A: Let C be the quarter circle centered at (2,0) that goes from (1,0) to (2,1).

Q: SHOW COMPLETE SOLUTION. PLEASE MAKE THE SOLUTION/HANDWRITING CLEAR. THANKYOU!

Q: w=f(z)=z/(1-z) under transformation IzI=3 Find the image of the circle?

Q: Consider the line integral given by , y-dx+xdy Ca) Evaluate the line integral where Cis the /ne…

A: We will find out the required value.

Q: ra a²-y² L". Sve²=v² a²-x²-y² . (x²z+y²z + z³) dzdxdy a²-x²-y²

Q: Evaluate the line integral, where C is the given curve. (x + 9y) dx + x² dy, C consists of line…

A: To evaluate the line integral, consider the curveC1 consisting of the line segment from0,0to9,1. The…

Q: Find extrema of f (x, y) = 2x² + 4y² on the unit-circle x + y² = 1.

A: To find the extrema of the function ƒ(x, y) = 2x² + 4y² on the unit-circle x² + y² = 1, we need to…

Q: Use hyperbolic functions to parametrize the intersection of the surfaces x' - y 49 and z= (Use…

Q: 4. (a) Sketch the curve r = 1 + sin 8, the cardioid curve. (b) Transform the equation of the curve r…

Q: Example Find the exact area of the surface obtained by rotating the curve y = v1+ ex, 0<x< 1, about…

Q: 6.(6 pts) Evaluate the line integral / (x+ y + 2) ds along the curve C given by r(t) = (sin t, cos…

Q: V4-x² e-z²-y² 12. Use a change to polar coordinates to evaluate the integral dy dr.

A: Answer

Question

**Problem Statement:**

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

\[
\oint_{C} xe^{-5x} \, dx + \left( x^4 + 2x^2y^2 \right) \, dy
\]

**Note:**
- $ C $ is the boundary of the region between the circles $ x^2 + y^2 = 9 $ and $ x^2 + y^2 = 25 $.

### Explanation:

**Green's Theorem** connects a line integral around a simple closed curve $ C $ to a double integral over the plane region $ D $ bounded by $ C $:

\[
\oint_{C} \left( M \, dx + N \, dy \right) = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA
\]

Here:

- $ M(x, y) = xe^{-5x} $
- $ N(x, y) = x^4 + 2x^2y^2 $

**Region $ C $**: The region is an annular area (the area between two circles) with radii determined by:

- The inner circle: $ x^2 + y^2 = 9 $, with radius 3.
- The outer circle: $ x^2 + y^2 = 25 $, with radius 5.

**Graphs/Diagrams:**

- Assume two concentric circles are drawn on a coordinate plane.
- The inner circle has a radius of 3 and the outer circle has a radius of 5.
- The region $ C $ is the shaded area between these two circles.

To solve, apply Green's Theorem to convert the line integral into a double integral over the annular region. Calculate $ \frac{\partial N}{\partial x} $ and $ \frac{\partial M}{\partial y} $, and integrate over the given region to find the area.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Double Integration$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Find an equation of the tangent plane to the surface at the given point. X²+y²+z²=14, (1,2,3) And find a set of symmetric equation for the normal line to the surface at the given point. ○X-1/1 = y-2/2 = z-3/3 ○X-1/14 = y-2/14 = z-3/14 ○X/1 = y/2 = z/3 ○X/14 = y/14 = z/14 ○X-1 = y-2 = z-3
Let F = 2(x + y)i + sin(y) 7. Find the line integral of Faround the perimeter of the rectangle with corners (4,0), (4,8), (-2, 8), (-2, 0), traversed in that order. line integral =
feni penc Yeni gizli Find the area of the region enclosed by the curve r = 2+ cos (20). Geçmiş İndirilenler Yer İşaretler (1.5) Yakınlaştır Yazdır.. (3,( Yayınla. Bul. Diğer araçlar Düzenle Ayarlar Yardım 9TT Çıkış 77
(a) Let f(z) = ---2i)(24 – 23) Using CRT, evaluate the integral Of(2)dz, where the contour C is the circle z-i+ 1| = 2.
Calculate using polar double integration the area of the region that lies between the circles r= 10000 sin (O) and r= 8391cos (0).
Let the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the integral ∫c F· dr = ∫c (2x-y)dx + (4y-x)dy in two different ways:(a) Parameterize the curve C and compute the integral directly. (b) Use the Fundamental Theorem of Line Integrals.
How do I change the function into polar coordinates?
Solve integral img1*when Γ is parametrically specified curve img2*const. a=7 Thanks
Use Green's theorem to evaluate the line integral I of the one-form W = + x sin? (y)) dx + (x² cos(y) sin(y)+ xy+sin" (y)) dy along the closed curve in R? formed by going from the origin to the point (T, 0) along the arc of the curve y = 8 sin(x), and then back to the origin along the x-axis.
1szs 9-x² - y² and x' +y <1 Sketch the surface H.
Consider the line integral I = rydy-y²dr where is the square cut from the first quadrant by the lines a 1 and y = 1. (i) Find the line integral (ii) Find the line integral using the circulation form of Green's Theorem. using the flux form of Green's Theorem.
QI Find the drea bounded by the two curves y=sinx and y=sin ax and the line x = =플 2