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Multivariable Calculus

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Difficult

Extrema Inertia

MVCALC-YX4KNG

Let a plane region $R$ be enclosed by parabola $y^2=x$ and line $x=1$, with constant density $\rho\equiv 1$.

Let $L$ be the family of lines pass through the origin, i.e., $x\sin\theta-y\cos\theta=0, \forall \theta$.

Find the maximal moment of inertial about the lines in $L$ of the region $R$.

A

$\cfrac{4}{3}$

B

$\cfrac{4}{15}$

C

$\cfrac{4}{7}$

D

$\cfrac{8}{15}$

E

$\cfrac{8}{7}$