Exercise · ZW-VECTOR-LINE-INTEGRAL-003 · ★☆☆ Basic
Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle x,y,z\rangle\) and \(\mathbf r(t)=\langle t,t,t\rangle\), \(0\le t\le2\).
Choose one option.
Show answer
- D. \(6\)
Explanation
\(\mathbf F(\mathbf r)\cdot\mathbf r'=\langle t,t,t\rangle\cdot\langle1,1,1\rangle=3t\) and \(\int_0^2 3t\,dt=6\).
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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