Question

Use a calculator or CAS to evaluate the line integral correct to four decimal places.
\[\int_c F\cdot dr\]
where
\[F(x,y)=xy\hat i+siny \hat j\]
and
\[\hat r(t)=e^t\hat i+e^{-t^2} \hat j\]
\[1\le t\le 2\]
I tried using this:
http://kevinmehall.net/p/equationexplorer/vectorfield
but it doesn't seem to be working.

Answer 1

I am gonna look at my old notes from Calc III

Answer 2

I can do it manually, but how do I do it using a calculator?

Answer 3

what's x and y?

Answer 4

\[\hat r '(t)=e^t\hat i-2e^{-t^2}\]
\[x=e^t\]
\]y=e^{-t^2}\]

Answer 5

http://www.wolframalpha.com/input/?i=Integrate%5BDot%5B%7Be%5Et*e%5E%28-t%5E2%29%2Csin%5Be%5E%28-t%5E2%29%5D%7D%2C%7Be%5Et%2Ce%5E%28-t%5E2%29%7D%5D%2C%7Bt%2C1%2C2%7D%5D

Answer 6

\[ y=e^{-t^2}\]

Answer 7

Ah i see. So I would set up the integral manually and then plug it into wolfram. Thank you!

Answer 8

well , I had it find dot product too

Answer 9

Turing, can I get a quick tutorial on how to plug this into wolfram?

Answer 10

Libniz seems to know better than I
I didn't know you could just write "dot" and get a dot product :P

Answer 11

I would have just done the dot product be hand, then written
"integrate ....dt from 1 to 2"

Answer 12

is it something along the lines of
integrate[Dot[{F( r(t))},{ r'(t)}],{t,1,2}]

Answer 13

I think that's right

Answer 14

ah libniz I do think you forgot to find r'

Answer 15

you don't need that when dealing with vector

Answer 16

but yeah, that seems to be how it's typed

Answer 17

http://www.wolframalpha.com/input/?i=Integrate%5BDot%5B%7Be%5Et*e%5E%28-t%5E2%29%2Csin%5Be%5E%28-t%5E2%29%5D%7D%2C%7Be%5Et%2C-2te%5E%28-t%5E2%29%7D%5D%2C%7Bt%2C1%2C2%7D%5D
right?

Answer 18

http://www.wolframalpha.com/input/?i=Integrate%5BDot%5B%7Be%5Et*e%5E%28-t%5E2%29%2Csin%5Be%5E%28-t%5E2%29%5D%7D%2CD%5B%7Be%5Et%2Ce%5E%28-t%5E2%29%7D%2Ct%5D%5D%2C%7Bt%2C1%2C2%7D%5D

Answer 19

\[\int\limits_C\vec F\cdot\vec r(t)dt=\int_a^b\vec F(\vec r(t))\cdot\vec r'(t)dt\]

Answer 20

indeed

Answer 21

that's what I would think @TuringTest

Answer 22

now Libniz and I agree
we just missed the r' earlier

Answer 23

oh yeah looks like you both came up with 1.9633

Answer 24

yup

Answer 25

Thanks y'all!