Exercise · ZW-VECTOR-LINE-INTEGRAL-006 · ★★☆ Intermediate
Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle xy,\,x\rangle\) and \(\mathbf r(t)=\langle t,t^2\rangle\), \(0\le t\le1\).
Choose one option.
A \(\frac{11}{12}\) B \(\frac{5}{4}\) C \(\frac{1}{4}\) D \(\frac{2}{3}\) Show answer ← Back to the question
Explanation \(\mathbf F(\mathbf r)=\langle t^3,t\rangle\), \(\mathbf r'=\langle1,2t\rangle\); the dot product is \(t^3+2t^2\) and its integral is \(\tfrac14+\tfrac23=\tfrac{11}{12}\).
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 \(C\).
\[\int_C \mathbf F\cdot d\mathbf r=\int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt\]
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