Question

~Line Integral question~

Answer 1

How do I calculate it when I am only given an f x ds? What is X? is x the function, or the value of a function?

Answer 2

is it f(x) = x, or an unknown function arbitrarily called x?

Answer 3

this looks simple, in the x-y plane

so for the first bit, you have y =x, so dy = dx

for the other parts, either dx or dy will be zero

Answer 4

Cleanup to avoid confusion. is the "x" in the integral x ds a function or the contents of a function? So is it g(x) = x --> Integral g(x) ds ?

Answer 5

the C on the integral means it is a closed loop...

so

Answer 6

there are a number of things that this can mean

for instance, the line from 0,0 to 1,1 means y = x, so dy = dx

Answer 7

I have no problem with that part - my part is understanding the integral x ds. I could probably solve it if they havent written than confusing integral x ds.

If I erase the line that says "integral x ds" I could figure out the functions of all three lines. Does integral x ds have any importance here, or is it just there to tell me that this is an line integral (scalar field? Conservative field?) ?

Answer 8

i think that is right

Answer 9

ds means scalar arc length

Answer 10

What does x symbolize? A coordinate or function that we have to parametrize ourselves?

Answer 11

x is x, ie the x coordinate in xy space

Answer 12

Sorry for not responding in a long time - Openstudy have issues that prevents the page from loading :(

Thank you for your help! I have received my answer. Making a new post for new questions instead of asking multiple questions in one post :)

Answer 13

No - in this context, \(x\) is the integrand and must therefore be parameterized. To be clear, I'll actually do the integral. The parameterizations are as follows - I'll diagram the first one for clarity.

Segment 1:
(0,0) \(\rightarrow\) (1,1)
Let \(x(s) = \frac{s}{\sqrt{2}}, s\in [0,\sqrt{2}]\)

Segment 2:
(1,1) \(\rightarrow\) (1,0)
Let \(x(s) = 1-s, s\in [0,1]\)

Segment 3:
(1,0) \(\rightarrow\) (0,0)
\(x(s) = 0\), so we can just forget this part