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A rod of length \( L \) is pivoted at an end. The linear mass density of the rod \( \lambda \) varies with the distance \( x \) from this end as \( \lambda=a x^{2}+b \mathrm{kgm}^{-1} \), where \( a \) and \( b \) are positive constants. Find the moment of inertia of the rod about the axis passing through this end and perpendicular to its length.
(a) \( \frac{2 a L^{5}}{5}+\frac{2 b L^{3}}{3} \)
(b) \( \frac{2 a L^{5}}{3}+\frac{2 b L^{3}}{5} \)
(c) \( \frac{a L^{5}}{7}+\frac{b L^{3}}{3} \)
(d) \( \frac{a L^{5}}{5}+\frac{b L^{3}}{3} \)
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