2009-11-09: Line Integrals
Stewart section 17.2
- Last time we introduced Vector Fields
- In section 17.2 we relate these to Line Integrals: an integral along a curve C.
- Game:
- Set up an arclength integral for C.
- If C is a wire and the density of the wire is d(x,y) = 3 + x + y g/cm, set up an integral for the mass of the wire.
- Set up an integral for the x center of mass.
- Key Point
- ds = sqrt(x'^{2} + y'^{2}) dt is the length of a tiny section of the curve.
- ds = |r'(t)| dt.
- Ths the integral int_{a}^{b} ds is the sum of these, that is, arclength.
- We can also sum any other property associated with the curve.
- Note that we also define the integrals with respect to x and y alone
- Game:
- Parameterize the line segment from (1,0) to (3,5).
- Integrate f(x,y) = x + 2y on this curve.
- Integrate F.dr on this curve, if F = <xy, 2y>.
- Key Point
- The integral of F.dr just adds up the component of F in the direction of C.
- But because the component of F in this direction is...
ma215-080-f09 lecture outline 2009-11-09
Created: Mon Nov 9 12:35:27 EST 2009
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©2009 Gavin LaRose