Line Integrals

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by stephaniejoy19
Last updated 5 years ago

Discipline:
Math
Subject:
Calculus

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Line Integrals

Line Integrals

When C is not made of straight line segments, it can be trickier to figure out how to parameterize the curve.

Here's what one looks like.

How do I parameterize the curve C?(when it is made of line segments)Well, we really like to use the interval 0 ≤ t ≤ 1. There is a trick that allows us to use this interval almost every time!In general, say C is a line segment from (a, b) to (c, d). Then, we can make parametric equations for C like this:x(t) = a + (c – a)ty(t) = b + (d – b)tSo, for example, if C is the line segment from (2, 5) to (0, 8), our parameterized curve would look like this:x(t) = 2 + (0 – 2)t = 2 – 2ty(t) = 5 + (8 – 5)t = 5 + 3tYou can check to make sure that your parametric equations work – plug in t = 0 and you should get the first point; plug in t = 1 and you should get the second point!

Sometimes, we have to evaluate line integrals over curves that do not consist of straight line segments. (Unfortunate, I know).

Make sure you read the problem carefully and take note of which of the three formulas to use! (Does the integral end in ds, dx, or dy?)

When possible, break C up into smaller pieces that are straight line segments and then add them together.For instance, if C is the boundary of a triangle, it can be split into 3 line segments, and in order to calculate the total line integral, you would just find the sum of the line integrals over each segment.


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