Q:

Find center of mass of triangle with varying density

My objective:

Find the center of mass of a thin triangular plate bounded by the y-axis and the lines  $y= 7x+3$ and $y= 36-4x$. Assume that the density is given by $\delta(x,y) = 
7x+2y+2$.

In class we were given the formula $\bar x = \frac{\iiint_V x\delta dV}{\iiint_V \delta dV}$ to find the x center, so I set it up like $\bar x = \frac{\int_{0}^{3}\int_{0}^{36} x(7x+2y+2)\,dydx}{\int_{0}^{3}\int_{0}^{36} (7x+2y+2)\,dydx} = \frac{156}{97}$, but that isn't correct. 
Is this approach wrong or did I make a mistake somewhere?

A:

I was taking the integral of a rectangle when I should have been taking it about the triangle.
As a result, the inner bounds should be changed and the the correct integral is $\bar x = \frac{\int_{0}^{3}\int_{7x+3}^{36-4x} x(7x+2y+2)\,dydx}{\int_{0}^{3}\int_{7x+3}^{36-4x} (7x+2y+2)\,dydx}$

