Change of Order of Integration

The ability to change the order of integration for Multiple Integral.

Example

${\int }_0^1{\int }_0^y(x^2-2xy+y^2)dxdy={\int }_0^1{\int }_x^1(x^2-2xy+y^2)dydx$

Theorem

$\int \int f(x,y)dA={\int }_a^b{\int }_{{\varphi }_1(x)}^{{\varphi }_2(x)}f(x,y)dydx={\int }_c^d{\int }_{{\psi }_1(y)}^{{\psi }_2(y)}f(x,y)dxdy$