- #1
asif zaidi
- 56
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Hi:
I am evaluating the curve integral below and I am getting an answer of 0. I have looked at my solution many times and cannot see that I have done anything wrong.
My concern is that a value of 0 for a curve integral does not make sense - a curve integral measures the distance from point A to point B on a curve so how can it be 0.
Problem Statement:
Consider the parametrized curve [tex]\varsigma[/tex]: [0:2[tex]\pi[/tex]] -> R[tex]^{s}[/tex], defined by [tex]\varsigma(t) [/tex] = (e[tex]^{t}[/tex]cos(t), e[tex]^{t}[/tex]sin(t)).
Evaluate the curve integral:
integral of ( (x / (x[tex]^{2}[/tex] + y[tex]^{2}[/tex]) ) dx + (y / (x[tex]^{2}[/tex] + y[tex]^{2}[/tex]) ) dy )
Problem Solution:
Step1: Calculate the norm of parametrized function
f(t) = e[tex]^{t}[/tex]cos(t) ; f'(t) = e[tex]^{t}[/tex]cos(t) - e[tex]^{t}[/tex]sin(t)
g(t) = e[tex]^{t}[/tex]sin(t) ; g'(t) = e[tex]^{t}[/tex]sin(t) + e[tex]^{t}[/tex]cos(t)
Therefore norm of || f'(t) + g'(t) || = sqrt(2) * e[tex]^{t}[/tex]
(I am not showing the intermediate steps)
Step2:
Evaluate f(x) at f(t), g(t) and g(x) at f(t), g(t)
f( f(t), g(t) ) = cos(t) / e[tex]^{t}[/tex]
g( f(t), g(t) ) = sin(t) / e[tex]^{t}[/tex]
Step3:
Multiply step2 and step 3 = cos(t) + sin(t)
Step4:
Evaluate integral of Step3
integral (0-2*pi) of (cos(t) + sin(t) dt ) = 0
Plz advise what I am doing incorrectly
Thanks
Asif
I am evaluating the curve integral below and I am getting an answer of 0. I have looked at my solution many times and cannot see that I have done anything wrong.
My concern is that a value of 0 for a curve integral does not make sense - a curve integral measures the distance from point A to point B on a curve so how can it be 0.
Problem Statement:
Consider the parametrized curve [tex]\varsigma[/tex]: [0:2[tex]\pi[/tex]] -> R[tex]^{s}[/tex], defined by [tex]\varsigma(t) [/tex] = (e[tex]^{t}[/tex]cos(t), e[tex]^{t}[/tex]sin(t)).
Evaluate the curve integral:
integral of ( (x / (x[tex]^{2}[/tex] + y[tex]^{2}[/tex]) ) dx + (y / (x[tex]^{2}[/tex] + y[tex]^{2}[/tex]) ) dy )
Problem Solution:
Step1: Calculate the norm of parametrized function
f(t) = e[tex]^{t}[/tex]cos(t) ; f'(t) = e[tex]^{t}[/tex]cos(t) - e[tex]^{t}[/tex]sin(t)
g(t) = e[tex]^{t}[/tex]sin(t) ; g'(t) = e[tex]^{t}[/tex]sin(t) + e[tex]^{t}[/tex]cos(t)
Therefore norm of || f'(t) + g'(t) || = sqrt(2) * e[tex]^{t}[/tex]
(I am not showing the intermediate steps)
Step2:
Evaluate f(x) at f(t), g(t) and g(x) at f(t), g(t)
f( f(t), g(t) ) = cos(t) / e[tex]^{t}[/tex]
g( f(t), g(t) ) = sin(t) / e[tex]^{t}[/tex]
Step3:
Multiply step2 and step 3 = cos(t) + sin(t)
Step4:
Evaluate integral of Step3
integral (0-2*pi) of (cos(t) + sin(t) dt ) = 0
Plz advise what I am doing incorrectly
Thanks
Asif