Find the line integral along the curve C from the origin along the x-axis to the

Question

Find the line integral along the curve C from the origin along the x-axis to the point (3, 0) and then counterclockwise around the circumference of the circle x^2 + y^2 = 9 to the point (3/squareroot 2, 3/squareroot 2). H vector = -y i vector + x j vector Find the line integral along the curve C from the origin along the x-axis to the point (6, 0) and then counterclockwise around the circumference of the circle x^2 + y^2 = 36 to the point (6/squareroot 2, 6/squareroot 2). F vector = y(x + 1)^-1 i vector + ln (x + 1)j vector

Explanation / Answer

1)given H=-yi +xj =<-y,x>

from origin to (3,0):

r(t)=<3t ,0>, 0<=t<=1

H(r(t))=<0,3t>

r'(t)=<3,0>

c1H.dr =[0 to 1]H(r(t)).r'(t) dt

=[0 to 1]<0,3t>.<3,0> dt

=[0 to 1]0 dt

=0

from (3,0) to (3/2 ,3/2):

r(t)=<3cost ,3sint>, 0<=t<=/4

H(r(t))=<-3sint,3cost >

r'(t)=<-3sint,3cost>

c2H.dr =[0 to /4]H(r(t)).r'(t) dt

=[0 to /4]<-3sint,3cost >.<-3sint,3cost> dt

=[0 to /4](9sin2t+9cos2t) dt

=[0 to /4](9) dt

=[0 to /4](9)t

=9(/4-0)

=9/4

cH.dr =c1H.dr+c2H.dr

cH.dr =0+(9/4)

cH.dr =9/4

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