Exercise · ZW-VECTOR-LINE-INTEGRAL-007 · ★★☆ Intermediate
Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle z,x,y\rangle\) and \(\mathbf r(t)=\langle t,t^2,t^3\rangle\), \(0\le t\le1\).
Choose one option.
A \(\frac{31}{60}\) B \(\frac{3}{2}\) C \(\frac{11}{12}\) D \(\frac{91}{60}\) Show answer ← Back to the question
Explanation \(\mathbf F(\mathbf r)=\langle t^3,t,t^2\rangle\) and \(\mathbf r'=\langle1,2t,3t^2\rangle\); the dot product is \(t^3+2t^2+3t^4\), with integral \(\tfrac14+\tfrac23+\tfrac35=\tfrac{91}{60}\).
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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