# 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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