Mass of a rectangle

Solution 1:

In two dimensions we have that $$\text{Mass}=\iint_R \delta(x,y)dxdy$$ In other words, mass is simply the integral of the density function over the region in question. Intuitively, we are continuously summing the mass distribution (i.e., the density) over the region in question, which gives the mass.

So now all you need is to find the limits of integration based on the boundary of the plate $R$. Notice the equations can be rewritten as $$y=0,$$ $$y=-\frac{b}{a}x+b,$$ $$y=-\frac{b}{c}x+b.$$ This describes a simple triangular region enclosed by these three equations. Since $c<a$ the third equation is the "lower" line. For a very helpful picture of what is going on, see an example here where I set $a=5,b=3,c=4$ (you can clearly see the region that is bounded in that example, its in the first quadrant by the way). Now we can set the limits of integration by observing that for each $y$ on the interval $[0,b]$, $x$ varies between (solving from the previous equations) $$x=-\frac{c}{b}y+c,$$ and $$x=-\frac{a}{b}y+a.$$ Finally we can write the integral as $$\text{Mass}=\iint_R \delta(x,y)dxdy=\int_0^b \int_{-\frac{c}{b}y+c}^{-\frac{a}{b}y+a} xdxdy.$$ You should now be able to evaluate the integral.