## 8

## Multiple Integrals

##### 8.1 DOUBLE INTEGRATION

Double integrals occur in many practical problems in science and engineering. It is used in problems involving area, volume, mass, centre of mass. In probability theory it is used to evaluate probabilities of two dimensional continuous random variables.

#### 8.1.1 Double Integrals in Cartesian Coordinates

A double integral is defined as the limit of a sum. Let *f*(*x*, *y*) be a continuous function of two independent variables *x* and *y* defined in a simple closed region R. Sub-divide R into element areas ΔA_{1}, ΔA_{2},…, ΔA_{n} by drawing lines parallel to the coordinate axes.

**Fig. 8.1**

Let (*x*_{i}, *y*_{i}) be any point in ΔA ...