Mr. Integral
Math notes/Calculus III/Double Integrals

Double Integrals and Accumulation — Cheat sheet

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Core rule

∬Rf dA=lim⁡∑f(xi,yi)ΔAi,\iint_Rf\,dA=\lim\sum f(x_i,y_i)\Delta A_i, fˉ=∬Rf dAarea⁡(R).\bar f=\frac{\iint_Rf\,dA}{\operatorname{area}(R)}.

Watch for

An inner integration treats the other variable as fixed. Signed accumulation, absolute volume and average value are different quantities.