Find the x-coordinate of the centroid of the region bounded by the following curves: y=x^2,y=x+6.
Hint: Make a sketch and use symmetry where possible.
x-coordinate of Centroid:
I got the wrong answer but I came up with 558/740.
I came up with a different answer, but we may start by finding the value of $\displaystyle x$ for which $\displaystyle y=x^2$ and $\displaystyle y=x+6$ intersect:
$\displaystyle \begin{aligned}
x^2&=x+6\\
x^2-x-6&=0\\
(x-3)(x+2)&=0.
\end{aligned}$
Our limits of integration are therefore $\displaystyle x=-2,3$. giving us the area of the region:
$\displaystyle A=\int_{-2}^3 (x+6)-x^2\,dx.$
To find the $\displaystyle x$-coordinate of the centroid, we just calculate
$\displaystyle \bar{x}=\frac{1}{A}\int_{-2}^3 x((x+6)-x^2)\,dx.$