meggiesmermaid
Evaluate the line integral
(x2+y2)dx+2xydy
where is the path of the semicircular arc of the circle x2+y2=64 starting at (8,0) and ending at (−8,0) going counterclockwise.
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TuringTest
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did you want to check an answer, or are you stuck?
meggiesmermaid
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\[512\int\limits_{0}^{\pi} \cos^2t sint - \sin^3t\] im stuck at how to integrate this.
TuringTest
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use\[u=\cos t\]and\[\sin^2t=1-\cos^2t\]
meggiesmermaid
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so du =-sinx
TuringTest
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how did you get cos^2t*sint for x^2+y^2dx ?
meggiesmermaid
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i keep getting confused but heres where i started \[\int\limits_{0}^{\pi} ((8cost)^2 + (8sint)^2)(-8sint) + 2(8cost)(8sint)(8cost)dt\]
TuringTest
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yes, and that can be written\[\int_0^\pi512(\sin^2t+\cos^2t)(-\sin t)dt+\int_0^\pi512(2\cos^2t\sin t) dt\]
meggiesmermaid
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\[-512\int\limits_{0}^{?\pi} sint dt +1024\int\limits_{0}^{\pi} \cos^2tsint\] which becomes this
TuringTest
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yes, which you can recombine to form this\[512\int_0^\pi2\cos^2t\sin t-\sin tdt\]
meggiesmermaid
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so would you use u=cost and du=-sint ?
TuringTest
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yep
meggiesmermaid
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or could you do \[sint(\cos^2t-1)\] which becomes \[\sin^3t\] ?
TuringTest
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and how to you propose to integrate that?
meggiesmermaid
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oh yeah.haha.
meggiesmermaid
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\[-512\int\limits_{1}^{-1}- 2u^2du - du \] ?
meggiesmermaid
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i mean + du
TuringTest
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yeah, looks good
TuringTest
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oh wait, how did 512 get negative?
meggiesmermaid
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im confused because du=-sint and the sint is positive so (-1/-1) would need to be multiplied right?
TuringTest
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yeah, but you already covered that by changing thethe signs og the terms
TuringTest
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however the double du notation is highly unorthodox and looks screwy to me, so I'd avoid it
TuringTest
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\[512\int_0^\pi2\cos^2t\sin t-\sin tdt\]\[u=\cos t\implies du=-\sin tdt\]\[512\int_{1}^{-1}- 2u^2du +\int_1^{-1} -\sin tdt\]
TuringTest
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sorry I messed up the bounds on the last integral, they should be 0 to pi
meggiesmermaid
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and the answer is 1706.66666666667?
meggiesmermaid
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nevermind thats wrong.
TuringTest
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\[512\int_0^\pi2\cos^2t\sin t-\sin tdt\]\[u=\cos t\implies du=-\sin tdt\]\[512\int_{1}^{-1}- 2u^2du +\int_0^{\pi} -\sin tdt\]let me see what I get...
TuringTest
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\[512\left(\left.-\frac23u^3\right|_1^{-1}+\left.\cos t\right|_0^\pi\right)\]\[512(\frac43-2)=512(-\frac23)\]but I always could have made a mistake...
meggiesmermaid
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its right! yay!
TuringTest
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sweet, I just hope you find your (likely algebra-based) mistake
meggiesmermaid
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thats what i got to too! thanks so much!
TuringTest
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welcome!