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Section 7.3 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{7.3.1} \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{7.3.2} \end{equation}

So how do you evaluate such an integral?

Use what you know!