RHIT MA113/Multiple Integral

Partial Derivatives Multiple Integral

Multiple Integral Edit

Evaluating Multiple Integrals Edit

Multiple 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 Edit

So 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 Edit

The Average value of a function   is equal to  

Areas/Volumes Edit

The 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 Edit

The 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

Center of Masses Edit