1.
Write out a parameterization for each of the two line segments mentioned above.
Section 1.3 Example Computing a Line Integral
While the previous video introduced the geometric idea of a line integral and derived a formula for it, we didn't actually do a computational example. Here we will find a parameterization for a given curve and plug everything into the formula
Learning Objectives:
- Find parameterizations for curves
- Evaluate Line Integrals for a given parameterization
Post-Video Activities Post-Video Activities
In the above video our curve was represented by one parameterization. However, sometimes a curve is easier to break into different segments and write out a parameterization for each segment. For instance, consider the curve that goes from the origin to (1,0) in a straight line and then up to (1,2) in a straight line. We have a property called Additivity where if a curve \(C_1\) ends at the start of a curve \(C_2\) then
\begin{gather*}
\int_{C_1\cup C_2}f(x,y)ds=\int_{C_1}f(x,y)ds+\int_{C_2}f(x,y)ds
\end{gather*}
2.
Compute \(\int_{C_1\cup C_2}(xy+x^2)ds\)
