Hallar el centroide (x ̅, y ̅ ) de la región limitada por la curva y = x ^ 2 y la rectay = x + 2?

Primero veamos cuáles son las abscisas de los puntos de intersección <img src="https://tex.z-dn.net/?f=x%5E2%3Dx%2B2%5C%5C%20%5C%5C%0Ax%5E2-x-2%3D0%5C%5C%20%5C%5C%0A%28x-2%29%28x%2B1%29%3D0%5C%5C%5C%5C%0A%5Cboxed%7Bx%3D-1%5Cvee%20x%3D-1%7D" />

Segundo hallemos el área de la región <img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0AA%3D%5Cint_%7B-1%7D%5E2x%2B2-x%5E2%5C%2Cdx%5C%5C%20%5C%5C%0AA%3D%5Cleft.%5Cleft%28%5Cfrac%7Bx%5E2%7D%7B2%7D%2B2x-%5Cdfrac%7Bx%5E3%7D%7B3%7D%5Cright%29%5Cright%7C_%7B-1%7D%5E2%5C%5C%20%5C%5C%0AA%3D%20%5Cleft%28%5Cfrac%7B2%5E2%7D%7B2%7D%2B2%282%29-%5Cdfrac%7B2%5E3%7D%7B3%7D%5Cright%29-%5Cleft%28%5Cfrac%7B%28-1%29%5E2%7D%7B2%7D%2B2%28-1%29-%5Cdfrac%7B%28-1%29%5E3%7D%7B3%7D%5Cright%29%5C%5C%20%5C%5C%0A%5Cboxed%7BA%3D%20%5Cdfrac%7B9%7D%7B2%7D%7D" />

Luego hallemos los momentos respecto de los ejes

EJE X

Sección transversal : <img src="https://tex.z-dn.net/?f=c%28x%29%3Dx%2B2-x%5E2" />

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0AM_x%3D%5Cint%20_%7B-1%7D%5E2x%5Ccdot%20c%28x%29%5C%2C%20dx%5C%5C%20%5C%5C%0AM_x%3D%5Cint%20_%7B-1%7D%5E2x%20%28x%2B2-x%5E2%29%5C%2C%20dx%5C%5C%20%5C%5C%0AM_x%3D%5Cint%20_%7B-1%7D%5E2x%5E2%2B2x-x%5E3%5C%2C%20dx%5C%5C%20%5C%5C%20%5C%5C%0A%5Cboxed%7BM_x%3D%5Cdfrac%7B9%7D%7B4%7D%7D" />

EJE Y

Sección transversal : <img src="https://tex.z-dn.net/?f=c_1%28y%29%3D2%5Csqrt%7By%7D%20%5C%3B%2C%5C%3B%20y%5Cin%20%5B0%2C1%5D" /> <img src="https://tex.z-dn.net/?f=c_2%28y%29%3D%5Csqrt%7By%7D%2B2-y%5C%3B%2C%5C%3B%20y%5Cin%20%5B1%2C4%5D" />

Entonces

<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%20%0AM_y%3D%5Cint_%7B0%7D%5E%7B4%7Dy%5Ccdot%20c%28y%29%5C%2Cdy%3D%5Cint_%7B0%7D%5E%7B1%7Dy%5Ccdot%20c_1%28y%29%5C%2Cdy%2B%5Cint_%7B1%7D%5E%7B2%7Dy%5Ccdot%20c_2%28y%29%5C%2Cdy%5C%5C%20%5C%5C%20%5C%5C%0AM_y%3D%5Cint_%7B0%7D%5E%7B1%7D2y%5Csqrt%7By%7D%5C%2C%20dy%20%2B%5Cint_%7B1%7D%5E%7B4%7Dy%5Csqrt%7By%7D%2B2y-y%5E2%5C%2Cdy%5C%5C%20%5C%5C%20%5C%5C%0A%5Cboxed%7BM_y%3D%5Cdfrac%7B36%7D%7B5%7D%7D%20" />

Finalmente los centros de masa

<img src="https://tex.z-dn.net/?f=%5Coverline%7Bx%7D%3D%5Cdfrac%7B9%2F4%7D%7B9%2F2%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5C%5C%20%5C%5C%20%5C%5C%20%5Coverline%7By%7D%3D%5Cdfrac%7B36%2F5%7D%7B9%2F2%7D%3D%5Cdfrac%7B8%7D%7B5%7D" />.