Maximizing a Line Integral

Calculus Level 5

Let \(r(t)=<f(t),g(t)>\) be a simple closed curve in 2-space for \(0 \leq t < T\), and \(r(0)=r(T)\).

It is known that \(\int_{0}^{T} \sqrt{[f'(t)]^{2}+[g'(t)]^{2}} \, dt=24\)

If the maximum value of \(\oint_{c}( x^{2}+8x ) \, dy + ( 2xy ) \, dx\) integrated along \(r(t)\) from \(0 \leq t < T\) is \(I\), find \(\lfloor I \rfloor\)

Details and Assumptions:

  • The region enclosed by \(r(t)\) from \(0 \leq t < T\) is simple, closed, piecewise smooth, and positively oriented in the \(xy\)-plane.
  • You may use that fact that \(\pi^{-1}\approx0.3183\).

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