Calculus/Multivariable Calculus/Double Integrals

Order of integration

In some cases the first integral of the entire iterated integral is difficult or impossible to solve, therefore, it can be to our advantage to change the order of integration.

$\int _{a}^{b}\,\int _{f(x)}^{g(x)}h(x,y)\,dxdy$

$\int _{c}^{d}\,\int _{e(y)}^{f(y)}h(x,y)\,dydx$

As of the writing of this, there is no set method to change an order of integration from dxdy to dydx or some other variable. Although, it is possible to change the order of integration in an x and y simple integration by simply switching the limits of integration around also, in non-simple x and y integrations the best method as of yet is to recreate the limits of the integration from the graph of the limits of integration.

In higher order integration that can't be graphed, the process can be very tedious. For example, dxdydz can be written into dzdydx, but first dxdydz must be switched to dydxdz and then to dydzdx and then to dzdydx (as 3-dimensional cases can be graphed, this method would lack parsimony).