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Limit for Double Integrals When Boundaries are Functions

Evaluation of double integrals is often simpple when the region of integration is rectangular eg  
\[0\leq x\leq 1, 2 \leq y\leq 3\]
. It is not so simple when one or more boundaries of the region of integration is given by an equation in the variables.
Suppose we are to evaluate  
\[\int_R xy dR\]
  over the region  
\[R\]
  satisfying  
\[x+2y \leq 4, x-3y \geq6, x \geq0\]
.

We can evaluate this integral by considering the limits. For the region  
\[R\]
   
\[x/3 -2 \leq y\leq2-x/2\]
  and  
\[0 \leq x\leq 24/5\]

Not that  
\[\frac{24}{5}\]
  is the  
\[x\]
  intecpt of the top and bottom boundaries of  
\[R\]
.
We can write the integral as
\[\begin{equation} \begin{aligned} \int^{24/5}_0 \int^{2-x/2}_{x/3-2} xy dy dx &= \int^{24/5}_0 [xy^2/2]^{2-x/2}_{x/3-2} dx \\ &= \int^{24/5}_0 ((x(2-x/2)^2 - x(x/3-2)^2) dx \\ &= \int^{24/5}_0 (-2x^2/3+5x^3/36) dx \\ &= [-2x^3/9 5x^4/144]^{24/5}_0 \\ &= 663/125 \end{aligned} \end{equation} \]

Notice that on the left hand boundary  
\[x=0 \]
  , a constant, and on the right hand boundary,  
\[x=24/5\]
.
This is why we integrated with respect to  
\[y\]
  first, then  
\[x\]
.