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\[\begin{equation} \begin{aligned} \int^1_0 \int^{\sqrt{1+x^2}}_0 y dydx &= \int^1_0 [ \frac{y^2}{2} ]^{\sqrt{1+x^2}}_0 dx \\&= \int^1_0 (\frac{1+x^2}{2}) dx \\ &= \int^1_0 \frac{1+x^2}{2} dx \\ &= [x+ \frac{x^3}{6} ]^1_0 \\ &= \frac{7}{6} \end{aligned} \end{equation} \]