Centroid Calculator

Find center of mass (centroid) and moments of a region/area step by step

The calculator will try to find the center of mass and moments 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.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please contact us.

Your Input

Find the center of mass of the region bounded by the curves $y = x^{2}$ , $y = 2 x$ .

Solution

$M_{x} = \int\limits_{0}^{2}\int\limits_{x^{2}}^{2 x} y\cdot 1\, dy\, dx = \frac{32}{15}\approx 2.133333333333333$

$M_{y} = \int\limits_{0}^{2}\int\limits_{x^{2}}^{2 x} x\cdot 1\, dy\, dx = \frac{4}{3}\approx 1.333333333333333$

$m = \int\limits_{0}^{2}\int\limits_{x^{2}}^{2 x} 1\, dy\, dx = \frac{4}{3}\approx 1.333333333333333$

$\left(\bar{x}, \bar{y}\right) = \left(\frac{M_{y}}{m}, \frac{M_{x}}{m}\right) = \left(1, \frac{8}{5}\right) = \left(1, 1.6\right)$