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:
1. Set up an arclength integral for C.
2. 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.
3. Set up an integral for the x center of mass.
• Key Point
1. ds = sqrt(x'2 + y'2) dt is the length of a tiny section of the curve.
2. ds = |r'(t)| dt.
3. Ths the integral intab ds is the sum of these, that is, arclength.
4. 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:
1. Parameterize the line segment from (1,0) to (3,5).
2. Integrate f(x,y) = x + 2y on this curve.
3. Integrate F.dr on this curve, if F = <xy, 2y>.
• Key Point
1. The integral of F.dr just adds up the component of F in the direction of C.
2. 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