June 11th, 2017, 04:38 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 165 Thanks: 2  parameterization and Integral
One parameterization for the unit circle C is r(t) = <cos(2t), sin(2t)>, for 0 <= t <= pi. Set up but do not evaluate the line integral of f(x, y) = x^2 + y^2 + 3 along C, using the parameterization above. Answer But.. according to my calculation, I got 4 dt only My reasoning: x^2+y^2+3 = (sin(2t))^2 + (cos(2t))^2+3 = 1 + 3 = 4 Any idea where I got it wrong?? 
June 11th, 2017, 04:51 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,595 Thanks: 1493 
You got 4 dt only? With no integral limits?

June 11th, 2017, 05:05 PM  #3 
Senior Member Joined: Jan 2017 From: Toronto Posts: 165 Thanks: 2  
June 11th, 2017, 06:18 PM  #4 
Global Moderator Joined: Dec 2006 Posts: 18,595 Thanks: 1493 
Okay. By default, a line integral is with respect to s, not t. For this problem, ds/dt = 2, so the answer given (in terms of t) is twice the answer you found. Do you know the formula by which ds/dt can be found? 
June 11th, 2017, 06:20 PM  #5 
Senior Member Joined: Jan 2017 From: Toronto Posts: 165 Thanks: 2  You think I may have missed putting some vital info about the question?

June 11th, 2017, 11:12 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 18,595 Thanks: 1493 
You should previously have been shown a comparable example of a line integral. You have to obtain $\displaystyle \int_0^\pi \!\text{f}(x,y)\frac{ds}{dt}dt$, where $\displaystyle \frac{ds}{dt} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}$, which implies that $ds/dt = 2$ for this problem, and where $\text{f}(x, y) = x^2 + y^2 + 3 = \cos^2(2t) + \sin^2(2t) + 3$, which equals 4. 

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integral, parameterization 
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