The Simple Double Integral

When a function is a function of two variables, or when a function is to be integrated over a region. For example.  
  may describe the height above the  
  plane over the region  
\[1 \lt x \lt 2, \; 0 \lt y \lt 1\]
. The integral will give the total volume over the region.
\[V= \int^{x=2}_{x=1} \int^{y=1}_{y=0} xe^y dy dx= \int^2_1 [ \int^1_0 xe^y dy] dx\]

We perform the integral; with respect to  
\[\begin{equation} \begin{aligned} V &= \int^{x=2}_{x=1} \int^{y=1}_{y=0} xe^y dy dx \\ &= \int^{x=2}_{x=1} [ \int^{y=1}_{y=0} xe^y dy] dx \\ &= \int^2_1 [xe^y]^{y=1}_{y=0}dx =\int^{x=2}_{x=1} x(e-1) dx \\ &= [ \frac{x^2}{2}(e-1)]^{x=2}_{x=1} \\ &= ( \frac{2^2}{2}- \frac{1^2}{2} ) (e-1) \\ &= \frac{3}{2}(e-1) \end{aligned} \end{equation}\]

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