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Function of 1 variable $$\int_a^b f(x)\,dx$$ = area below graph of f over [a,b];

Function of 2 variables $$\iint \limits_R f(x,y)\,dx\,dy $$ = volume of z=f(x,y) over a region R in XY plane

Example 1:

$$z=1-x^2-y^2$$

Region 0≤x≤1; 0≤y≤1

$$z=1-x^2-y^2$$

$$\iint \limits_R f(x,y)\,dx\,dy $$

$$=\int_0^1 \!\!\!\int_0^1 (1-x^2-y^2)\,dx\,dy$$

Example 2:

$$z=1-x^2-y^2$$

Region $$x^2+y^2=1$$, x≥0, y≥0