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## How To Compute Iterated IntegralsNow that we know what double integrals In order to integrate over a rectangle $[a,b] \times [c,d]$, we
first integrate with respect to one variable (say, $y$) for each
fixed value of $x$. That's an ordinary integral, which we
can compute using the fundamental theorem of calculus. We
then integrate the result over the other variable (in this case
$x$), which we can also compute using the fundamental theorem of
calculus. So a There are two ways to see the relation between double integrals
and iterated integrals. In the bottom-up approach, we
evaluate the sum $$\sum_{i=1}^m\sum_{j=1}^n
f\left(x_{i}^*,y_{j}\right)^* \,\Delta x\,\Delta y=\sum_{i=1}^m
\left(\sum_{j=1}^n f\left(x_{i}^*,y_{j}^*\right) \,\Delta
x\,\right) \Delta y,$$ by first summing over all of the boxes with
a fixed $i$ to get the contribution of a column (as indicated with
the parentheses on the second sum), and then adding up the
columns. (We could do this in the other order, by reversing
the summations.)
This approach is explained in the following video, and an example
is worked out. ( |