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# 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$$

![](/files/-Lf5hp2dzmcsEPm5sNLu)