Vector Line Integrals

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Section 10.3 Vector Line Integrals

Consider now the problem of finding the work $W$ done by a force $\FF$ in moving a particle along a curve $C\text{.}$ We begin with the relationship

\begin{equation} \hbox{work} = \hbox{force} \times \hbox{distance} .\tag{10.3.1} \end{equation}

Suppose you take a small step $d\rr$ along the curve. How much work was done? Since only the component along the curve matters, we need to take the dot product of $\FF$ with $d\rr\text{.}$ Adding this up along the curve yields

\begin{equation} W = \Lint \FF\cdot d\rr .\tag{10.3.2} \end{equation}

So how do you evaluate such an integral?

Use what you know!