Exercise · ZW-VECTOR-LINE-INTEGRAL-009 · ★★★ Advanced
The same curve \(C\) is traversed with the same orientation, but with a parametrization twice as fast. The line integral of \(\mathbf F\)…
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- C. stays the same
Explanation
The integral depends on the curve and its orientation, not on the speed: substituting \(t=2u\), \(\mathbf r'\,dt\) becomes \(d\mathbf r\) with no change in value.
Review the concept: Line integral of a vector field
The line integral of a vector field \(\mathbf F\) over an oriented curve \(C\), parametrized by \(\mathbf r(t)\), \(a\le t\le b\), measures the work \(\mathbf F\) does along \(C\).
\[\int_C \mathbf F\cdot d\mathbf r=\int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt\]
More practice problems: line integral of a vector field
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