Multiple integral

The multiple integral is a definite integral of a function of more than one real variable, for example, $f(x, y)$ or $f(x, y, z)$.

Integrals of a function of two variables over a region in $R^2$ are called double integrals, and integrals of a function of three variables over a region of $R^3$ are called triple integrals.

Here we only talk about double integrals.

1. Thinking an object which upper body is an wave, and its lower body is an squre. The x axis face you. The wave function is $f(x, y)$

2. Now put your knife down right through its center, what you will get is a slice of bread

3. We should calculates the area of that slice: $\int _\text{front} ^\text{back} f(x, y) \cdot dy$, undefined(so many) lines come together with y direction to a slice

4. Then undefined(so many) slices come together with x direction to an complete object: $\int _\text{left} ^\text{right} [\int _\text{front} ^\text{back} f(x, y) \cdot dy] \cdot dx$