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.

Density

\rho{\left(x,y \right)}

Curves:

Comma-separated. x-axis is

y = 0

, y-axis is

x = 0

Lower limit:

Optional.

Upper limit:

Optional.

If you are using periodic functions and the calculator cannot find a solution, try to specify the limits. If you don't know the exact limits, specify wider limits that contain the region (see example). Use the graphing calculator to determine the limits.

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Your Input

Find the moments of inertia of the region bounded by the curves $y = 3 x$ , $y = x^{2}$ .

Solution

$I_{x} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} y^{2}\cdot 1\, dy\, dx = \frac{2187}{28}\approx 78.107142857142857$

$I_{y} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} x^{2}\cdot 1\, dy\, dx = \frac{243}{20} = 12.15$

$m = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} 1\, dy\, dx = \frac{9}{2} = 4.5$

$R_{x} = \sqrt{\frac{I_{x}}{m}} = \frac{9 \sqrt{42}}{14}\approx 4.166190448976482$

$R_{y} = \sqrt{\frac{I_{y}}{m}} = \frac{3 \sqrt{30}}{10}\approx 1.643167672515498$