{cal3 exam3} Line integrals

Steps for a line $\int$ wrt. x or y on line segment

1. @ param some $f(x,y)=f(x(t))$
2. Find $r(t)$ like in previous step
3. Our Jacobian here is simply $x^{\prime }$ or $y^{\prime }$

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Steps for a line $\int$ wrt. arc length

lamredu

1. Find r(t) denoted by a parametric EQ
   1. In case of a line segment its form takes $r(t)=(1-t)<x_0,y_0\cdots >+t<x_1,y_1,\cdots >$ for $0\le t\le 1$
2. Find $||r^{\prime }(t)||$
3. Transform the insides to match with whatever $x=..$ $y=..$
4. Plug into the general form

\int\limits_{C} f(x, y) \, ds = \int\limits_{a}^{b} f(h(t), g(t)) \, \| \vec{r}\,'(t) \| \, dt

Example Q for line \int!

• $f(x)=x^{2}y+x$, from $(1,-1),(-4,1)$-
• -m @param line, get $r(t)=<-5t+1,2t-1>$, then find $|r^{\prime }(t)|||$
• Transform insides given $x=-5t+1,y=2t-1$
• Solve where a=0,b=0 (by )

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(todo, need to read over a little first..)

• DIvergence
• Graident
• Curl

Q1 Let R be filled triangle w/ verticies $(1,1),(-1,1),(0,0)$