Exercise · ZW-VECTOR-LINE-INTEGRAL-010 · ★★★ Advanced
Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle 2xy,\,x^2\rangle\) and \(\mathbf r(t)=\langle\cos t,\sin t\rangle\), \(0\le t\le\tfrac\pi2\).
Choose one option.
A \(0\) B \(- \frac{2}{3}\) C \(\frac{2}{3}\) D \(\frac{4}{3}\) Show answer ← Back to the question
Explanation The dot product is \(-2\cos t\sin^2t+\cos^3t\). Its integrals from \(0\) to \(\tfrac\pi2\) are \(-\tfrac23\) and \(\tfrac23\), which cancel. (It also follows from \(x^2y\) being \(0\) at both endpoints.)
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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