Applied Mathematics 2277A/B Chapter 15.3: Applied Mathematics 2277A/B Chapter 15.: Textbook 15.3
13 Feb 2019
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Line Integrals
Introduction
Single integrals can be practically visualized as area b/w the x-axis and the f(x) bounded by x points a
and b
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Double integrals can be practically visualized as the area of a 2D region in space
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Triple integrals can be practically visualized as the volume of a 3D region in space
-
Recall:
It may happen that we are looking for the area of a region above a curve instead of just the x-axis.
It is in these cases that we need line integrals.
Line Integrals
Def: A line integral is an integral where the axis of integration is a curve.
Where c is the curve, and s is the length of the curve (because we can't just do dx because that therefore
implies the length of the entire x axis)
Solving Line Integrals
From a derivation in earlier chapters, we learned that the length of C can be evaluated by expressing the arc
length element ds in terms of a infinitesimalization of a position vector r and the infinitesimalization of the
parameter upon which the position vector depends.
So therefore we can establish a way of solving line integrals by substituting this in
Example 1
(there is another case where we may need to
take find areal density over a surface, which
is again another nope. That's called a surface
integral. We might look at that later)
where r = r(t), (a < t <b)
Textbook 15.3
February 4, 2019
9:17 PM
Lecture Notes Page 1
Document Summary
Single integrals can be practically visualized as area b/w the x-axis and the f(x) bounded by x points a and b. Double integrals can be practically visualized as the area of a 2d region in space. Triple integrals can be practically visualized as the volume of a 3d region in space. It may happen that we are looking for the area of a region above a curve instead of just the x-axis. It is in these cases that we need line integrals. (there is another case where we may need to take find areal density over a surface, which is again another nope. Def: a line integral is an integral where the axis of integration is a curve. Where c is the curve, and s is the length of the curve (because we can"t just do dx because that therefore implies the length of the entire x axis)
