Line integral over half circle.. I'm stuck!

Oct 2012
5
0
Minneapolis, MN
Hello ppl!

So I'm frustrated with this problem:Evaluate the line integral
c 5xy2ds
, where C is the right half of the circle x2+y​2=36

So far, I've taken x=6Cos(t) and y = 6Sin(t) to parametrize a circle with radius of 6, then taken their modulus (to get rid of ds and get dt) and got 6 because rest of it cancels out..
After that I replaced the xy2 with the trig. values to get 5*6*Cos(t)Sin2(t). After that when I integrate it with respect to t, 3Pi/2 < t < Pi/2 to go over only the right side of the circle, I'm getting 20, which is wrong because the browser is rejecting it as the wrong answer.. :/
I think I'm doing something really silly. Anyone wanna help?

Thanks!
-Sparrow
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
Hello ppl!

So I'm frustrated with this problem:Evaluate the line integral
c 5xy2ds
, where C is the right half of the circle x2+y​2=36

So far, I've taken x=6Cos(t) and y = 6Sin(t) to parametrize a circle with radius of 6, then taken their modulus (to get rid of ds and get dt) and got 6 because rest of it cancels out..
After that I replaced the xy2 with the trig. values to get 5*6*Cos(t)Sin2(t). After that when I integrate it with respect to t, 3Pi/2 < t < Pi/2 to go over only the right side of the circle, I'm getting 20, which is wrong because the browser is rejecting it as the wrong answer.. :/
I think I'm doing something really silly. Anyone wanna help?

Thanks!
-Sparrow
The region \(\displaystyle \displaystyle \begin{align*} \frac{\pi}{2} < t < \frac{3\pi}{2} \end{align*}\) is actually the LEFT side of the circle. You should use \(\displaystyle \displaystyle \begin{align*} -\frac{\pi}{2} < t < \frac{\pi}{2} \end{align*}\)
 
Oct 2012
5
0
Minneapolis, MN
Thanks for the help but that just gives me 20 again! :p
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
I think you'll find \(\displaystyle \displaystyle \begin{align*} 5\,x\,y^2 = 5 \left( 6\cos{t} \right) \left( 6\sin{t} \right)^2 = 30\cos{t} \cdot 36\sin^2{t} = 720\cos{t}\sin^2{t} \end{align*}\)
 
Oct 2012
5
0
Minneapolis, MN
Gosh! totally screwed that one didn't I? Thankfully I have unlimited tries for this one...
Okay So I tried it with 720 Cos(t)Sin^2(t), but still the wrong answer..
Also, (at the risk of sounding stupid), I did it with 1080Cos(t)Sin^2(t) because 30*36 = 1080..
But still no luck.. Thank you so much for your help though..
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
Gosh! totally screwed that one didn't I? Thankfully I have unlimited tries for this one...
Okay So I tried it with 720 Cos(t)Sin^2(t), but still the wrong answer..
Also, (at the risk of sounding stupid), I did it with 1080Cos(t)Sin^2(t) because 30*36 = 1080..
But still no luck.. Thank you so much for your help though..
Yes I really should learn to multiply.

Did you remember to times this by 6 from your ds?
 
Oct 2012
5
0
Minneapolis, MN
DANG! That's what was wrong! Lot's of n00b mistakes in here on my part... Gosh thank you so much for your help, kind person!
Saved me a night's sleep you did! :)