Double integrals is useful to find the area of the region bounded in a co-ordinate plane.

**1. Area of the region** f(x,y) = c is given by the formula Area = $\int$ $\int$ dx . dy

In polar form, **the area is given by $\int$ $\int$ r dr. d$\theta$**

**2**. When the curve is given by z = f( x, y)

the s**urface area of the solid i**s given by** $\int$ $\int$ $\sqrt{\left ( \frac{\partial z}{\partial x} \right )^{2}+\left ( \frac{\partial z}{\partial y} \right )^{2}+1}$ . dx. dy**

Apart from evaluating the area and the volume we can also compute mass, electric charge, center of mass and moment of inertia.

**3**. **Mass of a Lamina** whose density function is given by $\rho$ ( x, y ) can be calculated using the formula, **mass ( m ) = $\iint_{D}^{}$ $\rho$ ( x, y ) dx. dy.**

**4.** If an **electric charge** is distributed over a region D and the charge density (in units of charge per unit area) is given by $\sigma$ (x, y) at a point (x, y) in D,

then **the total charge Q is = $\iint_{D}^{}$ $\sigma$ ( x, y ) dx. dy**

**5**. If the density function of a lamina is given by $\rho$ (x, y), then **the moment of the entire lamina** about the x and y axis are denoted by M_{x} and M_{y} respectively.

The double integral formula is given by,

**M**_{x} = $\iint_{D}^{}$ y . $\rho$ ( x, y ) . dx. dy

and** M**_{y} = $\iint_{D}^{}$ x. $\rho$ ( x, y ) . dx. dy.

**6.** The **center of mass of a lamina i**s given by the co-ordinates, ($\overline{x }$, $\overline{y }$).

When the lamina of density function $\rho$ (x, y) occupy the region D, then the co-ordinates are given by,

** $\overline{x }$ = $\frac{M_{y}}{m}$ = $\frac{1}{m}$ $\iint_{D}^{}$ y. $\rho$ ( x, y ) . dx. dy**

** $\overline{y }$ = $\frac{M_{x}}{m}$ = $\frac{1}{m}$ $\iint_{D}^{}$ x. $\rho$ ( x, y ) . dx. dy**

where m is given by** m = = $\iint_{D}^{}$ $\rho$ (x, y) dx. dy.**

**7**. **The moment of inertia or the second moment of a lamina **of mass m whose density function is given by $\rho$ (x, y) bounded by the region D can be calculated using the following integrals.

**Moment of Inertia about the x-axis = I**_{x} = $\iint_{D}^{}$ y^{2} . $\rho$ ( x, y ) dx dy

** Moment of Inertia about the y-axis = I**_{y} = $\iint_{D}^{}$ x^{2} . $\rho$ ( x, y ) dx dy

**Moment of Inertia about the Origin = Polar Moment of Inertia**

** = $\iint_{D}^{}$ ( x**^{2 }+ y^{2} ) . $\rho$ ( x, y ) dx dy

** and I**_{o} = I_{x} + I_{y}