Vector fields and line integrals - plane curves
Integrating a vector field along a curve is useful for computations such as work and circulation. Suppose is a vector field and the points of a curve lie in the domain of . Let be the unit tangent vector to . The line integral of along is
.
If is parametrized by for , then it turns out that
,
which is useful for actual computation.
In this interactive figure, carefully work down the controls in the right-hand pane from top to bottom, trying the sliders and checkboxes to see their effect. See how the parametrization of and the definition of are reflected in the graph. By examining and , you should be able to see which points on make the integrand positive, and which make it negative. From this you may gain some intuition about whether the line integral of along should be positive or negative.
Developed for use with Thomas' Calculus and Interactive Calculus, published by Pearson.