Moments of Inertia Calculator
Find moments of inertia and radii of giration of a region/area step by step
The calculator will try to find the moments of inertia and radii of gyration of the region/area bounded by the given curves, with steps shown.
Your Input
Find the moments of inertia of the region bounded by the curves $$$y = 3 x$$$, $$$y = x^{2}$$$.
Solution
$$$I_{x} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} y^{2}\cdot 1\, dy\, dx = \frac{2187}{28}\approx 78.107142857142857$$$
$$$I_{y} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} x^{2}\cdot 1\, dy\, dx = \frac{243}{20} = 12.15$$$
$$$m = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} 1\, dy\, dx = \frac{9}{2} = 4.5$$$
$$$R_{x} = \sqrt{\frac{I_{x}}{m}} = \frac{9 \sqrt{42}}{14}\approx 4.166190448976482$$$
$$$R_{y} = \sqrt{\frac{I_{y}}{m}} = \frac{3 \sqrt{30}}{10}\approx 1.643167672515498$$$
