# RHIT MA113/Multiple Integral

# Multiple Integral

edit## Evaluating Multiple Integrals

editMultiple Integrals are evaluated from the inside out, beginning by evaluating the innermost integral, then working outwards.

The inner integrals may have limits containing variables, so long as those variables are integrated in an enclosing integral. Because of this, the limits of outermost integrals must contain only constants.

## Changing the Order of Integration

editSo long as the order of integration is changed correctly, the multiple integral will cover the same region, and therefore order will not affect the end result of the multiple integral. In general, it is wise to begin by establishing the limits of the outermost integral first, then working inwards, to avoid any mistakes.

## Converting Coordinate Systems

edit### Cartesian to Cylindrical

edit### Cartesian to Spherical

edit### Cylindrical to Spherical

edit## Uses

edit### Average Value

editThe Average value of a function is equal to

### Areas/Volumes

editThe equation for Area is and Volume is

In Cartesian coordinates, and , therefore Area and Volume are and

The same process can be used in Polar, Cylindrical, and Spherical coordinates, as follows:

In Polar,

In Cylindrical,

In Spherical,

### Masses

editThe equation for the mass of an object is , where is the density of the object (which could be either a constant or function of position)

### Moments

edit#### First Moments

edit, where r is the distance from the axis or line of rotation

#### Second Moments

edit, where r is the distance from the axis or line of rotation