Maximizing a Line Integral

Green's Theorem helps us evaluate the line integral, call it $L$ :

$L=\oint_{c} \left( x^{2}+8x \right) dy + \left( 2xy \right) dx= \iint _{ R } \left( { \frac { \partial }{ \partial x } x^{ 2 }+8x }-\frac { \partial }{ \partial y } 2xy \right) dA = 8 \iint_{R} dA$

Where $R$ is the region enclosed by $r(t)$ , $A_{enc}$ .

In essence, $L$ is simply $8A_{enc}$ .

Our other integral is simply the arclength formula for a parametric function, so we know that the perimeter of whatever region $r(t)$ encloses will be $24$ .

It is well known that the shape in 2-space that maximizes area for a given perimeter is a circle, therefore $R$ must be a circle of circumference $24$ .

$2\pi r=24 \rightarrow A_{enc}=\frac{144}{\pi} \rightarrow I=8A_{enc}=\frac{1152}{\pi} \approx 366.69$

And so $\lfloor I \rfloor=\boxed{366}$