Question

Find the centroid (x¯,y¯) of the region bounded by:

y=3x^2+3x, y=0, x=0, and x=4

Answer 1

https://i.gyazo.com/34d3aa81faa7d27ef7750b8308bb53a7.png

\[\bar x = \dfrac{\int_{0}^{4} \; dx \quad xy}{\int_{0}^{4} \; dx \quad
y }\]

Answer 2

or \[x =\frac{ 1 }{ A } \int\limits_{0}^{4} xf(x) dx\] and \[y = \frac{ 1 }{ A } \int\limits_{0}^{4} \frac{ 1 }{ 2 } (f(x))^{2}\]

where \[A = \int\limits_{0}^{4} f(x) dx\]