## 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
$y$
first, with
$x \lt y \lt x^2$
the with respect to
$x$
, 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
$R$
in the
$xy$
plane is bounded by the curve
$y^2-x^2=1$
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{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}