Iterated Integration

An iterated integration is a double or triple integration such that the limits are functions of the variables in subsequent integrations.
For example  
\[\int^2_1 \int^{x^2}_x y dy dx\]
  is an iterated integration. Integrate with respect to  
  first, with  
\[x \lt y \lt x^2\]
  the with respect to  
, with  
\[1 \lt x \lt 2\]
Iterated integration is used with integrating over a region bound by surfaces which are functions of coordinates.
Example: The region  
  in the  
  plane is bounded by the curve  
  and the values  
\[0 \lt x \lt 1, 0 \lt y\]
. Find the value of  
\[\int_R y \]
We can write  
\[0 \lt x \lt 1, \; 0 \lt y \lt \sqrt{1+x^2}\]
. The integral becomes
\[\begin{equation} \begin{aligned} \int^1_0 \int^{\sqrt{1+x^2}}_0 y dydx &= \int^1_0 [ \frac{y^2}{2} ]^{\sqrt{1+x^2}}_0 dx \\&= \int^1_0 (\frac{1+x^2}{2}) dx \\ &= \int^1_0 \frac{1+x^2}{2} dx \\ &= [x+ \frac{x^3}{6} ]^1_0 \\ &= \frac{7}{6} \end{aligned} \end{equation} \]

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