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.

Comma-separated. x-axis is y=0y = 0, y-axis is x=0x = 0.
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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=3xy = 3 x, y=x2y = x^{2}.

Solution

Ix=03x23xy21dydx=21872878.107142857142857I_{x} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} y^{2}\cdot 1\, dy\, dx = \frac{2187}{28}\approx 78.107142857142857

Iy=03x23xx21dydx=24320=12.15I_{y} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} x^{2}\cdot 1\, dy\, dx = \frac{243}{20} = 12.15

m=03x23x1dydx=92=4.5m = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} 1\, dy\, dx = \frac{9}{2} = 4.5

Rx=Ixm=942144.166190448976482R_{x} = \sqrt{\frac{I_{x}}{m}} = \frac{9 \sqrt{42}}{14}\approx 4.166190448976482

Ry=Iym=330101.643167672515498R_{y} = \sqrt{\frac{I_{y}}{m}} = \frac{3 \sqrt{30}}{10}\approx 1.643167672515498

Region bounded by y = 3*x, y = x^2