CF - dr, where C is the intersection of the surface z= e-x2+y2 and the cylinder x2+y2=9 and where

F(x,y,z)=ln(2x+1)i+2y+1j+xeyk

Short Answer

Expert verified

As a result, the required integral is CF(x,y,z)×dr=36π.

Step by step solution

01

Step 1:Given data

F(x,y,z)=3x+3,4x+lny2+1z,2x+y

The goal is to calculate the line integral.CF(x,y,z)×dr, where the curve C$is defined as follows:

The surface is intersected by curve C.z=e-x2+y2x2+y2=9

02

Step2:Stokes theorem

To calculate this integral, use Stokes' Theorem. According to Stokes' Theorem,

"Assume S is an oriented, smooth or piecewise-smooth surface enclosed by a curve C. Assume n is an oriented unit normal vector of S and C has a parametrization that traverses C counterclockwise with respect to n.

If a vector fieldF(x,y,z)=F1(x,y,z)i+F2(x,y,z)j+F3(x,y,z)kis defined on S, then,

localid="1650793402578" CF(x,y,z)×dr=ScurlF(x,y,z)×ndS

03

Find Curl vector

First, determine the vector field's curl. F(x,y,z)=3x+3,4x+lny2+1-z,2x+y

First, determine the vector field's curl.F(x,y,z)=F1(x,y,z)i+F2(x,y,z)j+F3(x,y,z)kis defined as follows:

curlF(x,y,z)=ijkxyzF1(x,y,z)F2(x,y,z)F3(x,y,z)

=F3yF2ziF3xF1zj+F2xF1yk

First, determine the vector field's curl. F(x,y,z)=3x+3,4x+lny2+1-z,2x+ywill be,

curlF(x,y,z)=ijkxyz3x+34x+lny2+1z2x+y

=y(2x+y)z4x+lny2+1zix(2x+y)z(3x+3)j

+x4x+lny2+1zy(3x+3)k

=[1(1)]i[20]j+[40]k

=2i-2j+4k

==2,2,4.

04

Step 4:Vector perpendicular surface

If z=z(x,y), then the following vector;

n=z^x,z^y,1

is normal to the surface

Now, for z=e-x2+y2,It produces the vector perpendicular to this surface shown below. ;

n=z^x,^y,1

=xex2+y2,yex2+y2,1

=2xex2+y2,2yex2+y2,1.

05

Step5:The value of curl

Then, the value of curlF(x,y,z)- nwill be,

role="math" localid="1650791535292" curlF(x,y,z)×n=2,2,4×2xex2+y2,2yex2+y2,1

=(2)×2xex2+y2+(2)×2yex2+y2+(4)×(1)

=4xex2+y24yex2+y2+4

=4(xy)ex2+y2+4

06

Step 6:described the circle

The surface is intersected by curve C. z=e-x2+y2and the cylinder x2+y2=9, as a result, integration zone D is the circle-described disk. x2+y2=9in the xy-plane.

The region of integration is described in polar coordinates as follows.,

D={(r,θ)0r3,0θ2π}.

x=rcosθ,y=rsinθ,x2+y2=r2

and

dA=rdrdθ

07

Step 7:Applying stokes theorem

Now, apply Stokes' Theorem (1) to determine the integral.F(x,y,z)×dras follows:

F(x,y,z)×dr

=ScurlF(x,y,z)×ndS

=D4(xy)ex2+y2+4dA

=02π034(rcosθrsinθ)er2+4rdrdθ

=02π034rer2(cosθsinθ)+4rdrdθ

=02π034r2er2(cosθsinθ)+4rdrdθ

=0302π4r2er2(cosθsinθ)+4rdθdr

role="math" localid="1650793299338" =034r2er2(sinθ+cosθ)+4rθ02πdr

=02π034r2er2(cosθsinθ)+4rdrdθ

role="math" localid="1650791972329" =034r2er2(sin2π+cos2π)+4r×2π4r2er2(sin0+cos0)+4r×0dr

=034r2er2(0+1)+8πr4r2er2(0+1)+0dr

=034r2er2(0+1)+8πr4r2er2(0+1)+0dr

=034r2er2+8πr4r2er2dr

=038πrdr

=4πr203

=4π(3)24π(0)2

=36π

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 Vaia!

One App. One Place for Learning.

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

Get started for free

Study anywhere. Anytime. Across all devices.

Sign-up for free