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Line integral question

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Not sure how to go about answering this question and whether it requires parameterization. I need to evaluate the line integral of

(y-y^2)i + (x-2xy)j +k

where c is two line segments from (0,1) to (2,1) and from (2,1) to (2,3), picture above

(edited 6 years ago)

Reply 1

atsruser

Original post by BinaryJava

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Not sure how to go about answering this question and whether it requires parameterization. I need to evaluate the line integral of

(y-y^2)i + (x-2xy)j +k

where c is two line segments from (0,1) to (2,1) and from (2,1) to (2,3), picture above

This doesn't seem to need any parameterisation. You have a line of constant y followed by a line of constant x, so along the first line, you can substitute in the value of y and integrate against x, and vice versa for the second.

Reply 2

DFranklin

Original post by atsruser

But you need to know what the "d..." bit is, surely? If it's an integral w.r.t. ds, for example, this would be quite different from a ".dr" integral, for example.

Reply 3

atsruser

Original post by DFranklin

But you need to know what the "d..." bit is, surely? If it's an integral w.r.t. ds, for example, this would be quite different from a ".dr" integral, for example.

From the limited part of the question that I could see, I guessed that it's a

.d\bold{r}

Reply 4

BinaryJava

Original post by atsruser

From the limited part of the question that I could see, I guessed that it's a

.d\bold{r}

Yes its a dr, do you split it into two parts?

Reply 5

DFranklin

Original post by BinaryJava

Yes its a dr, do you split it into two parts?

Yes, you split it into 2 parts.

e.g. for the first part, we have r going from (0, 1) to (2, 1). For notation later, call this part of the curve

C_1

You ask if this requires parameterization; you can "kind of " avoid parameterization by using the x-coordinate itself as a parameterization, but until you are confident with what is going on, I would use a formal parameterization.

So parameterize with t; we want r = (0, 1) when t =0 and (2, 1) when t = 1, so we can write r = (2t, 1).

Then you can rewrite

\displaystyle \int_{C_1} {\bf F . dr}

\displaystyle\int_{t=0}^{t=1} {\bf F .} \dfrac{d{\bf r}}{dt}\, dt

, which will leave you with a standard (A-level style) integral.

Do the same thing for the 2nd part of the curve and then add them together.

Reply 6

BinaryJava

Original post by DFranklin

Yes, you split it into 2 parts.

e.g. for the first part, we have r going from (0, 1) to (2, 1). For notation later, call this part of the curve

C_1

\displaystyle \int_{C_1} {\bf F . dr}

\displaystyle\int_{t=0}^{t=1} {\bf F .} \dfrac{d{\bf r}}{dt}\, dt

, which will leave you with a standard (A-level style) integral.

Do the same thing for the 2nd part of the curve and then add them together.

I will give that a crack tomorrow, out of interest how would the non parametrization work?

Reply 7

atsruser

Original post by BinaryJava

I will give that a crack tomorrow, out of interest how would the non parametrization work?

You are integrating along lines of constant x and constant y, so you can set x and y to the appropriate constant and integrate against the other variable. For practice though I would agree that it's best just to parameterise.