Double Integration

Definition

We use single integrals to calculate the area and the arc length, and here, we use double integrals to calculate the solids.

Through the limit of the Riemann sum ${\displaystyle \underset{|p|\to 0}{lim}\sum _{k=1}^{n}f(x,y)\mathrm{\Delta }{A}_k}$, we say ${\displaystyle f}$ is integrable on ${\displaystyle R}$ if the limit exists. And the double integral of ${\displaystyle f}$ is ${\displaystyle {\iint }_{R}f(x,y)dA}$, which equals to the limit of the Riemann sum.

Integrability

Iterated Integral