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ch15guide - Math 241 Chapter 15 Integral Study Guide...

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Unformatted text preview: Math 241 Chapter 15 Integral Study Guide Important 1: Curves are always parametrized by ¯ r ( t ) = x ( t )ˆ ı + y ( t ) ˆ + z ( t ) ˆ k for a ≤ t ≤ b Note that some components might be 0 and the ˆ k component will definitely be 0 in 2D. Important 2: Surfaces are always parametrized by ¯ r ( u,v ) = x ( u,v )ˆ ı + y ( u,v ) ˆ + z ( u,v ) ˆ k for u,v restricted by R . Note that often it’s not u,v but x,y or z,θ or ... Important 3: Keep in mind that integraltext C 1 ds is length of C , integraltextintegraltext Σ 1 dS is surface area of Σ, integraltextintegraltext R 1 dA is area of R and integraltextintegraltextintegraltext D 1 dV is volume of D . 1. Line integral of a function. One choice: integraldisplay C f ds = integraldisplay b a f ( x ( t ) ,y ( t ) ,z ( t )) || ¯ r ′ ( t ) || dt 2. Line integral of a vector field. Lots of possibilities. This one has the most options....
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