So I am trying to find the mass of this rod which is a parallelogram. The link to the depiction of the rod is provided here: Desmos graph of rod. The mass of the rod given a density function $\rho (x,y)$, would be provided by :
$$\int_{y = -r\cos (\theta)}^{y = 0}\int_{x = -\tan (\theta)y}^{x = -\tan (\theta)y + s} \rho (x,y) dx dy$$
Where r is the length of the rod, $\theta$ is the angle from the vertical axis (the y-axis), and s is the width of the rod. I defined my coordinates such that downward is negative.
Let's say that r = 7, $\theta$ = $\frac{\pi }{6}$, and s = 2. Let's also say that, for some reason, the density function is defined as $\rho (x,y) = sin(x)cos(y)$. If you evaluate the double integral, you would get a negative result. What does that mean in this case? The mass cannot be negative.
No, a mass cannot be negative. But a density function cannot take negative values either.