RHIT MA113/Multiple Integral

Partial Derivatives Multiple Integral

Multiple IntegralEdit

Evaluating Multiple IntegralsEdit

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 IntegrationEdit

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 SystemsEdit

Cartesian to CylindricalEdit

Cartesian to SphericalEdit

Cylindrical to SphericalEdit


Average ValueEdit

The Average value of a function   is equal to  


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,  


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)


First MomentsEdit

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

Second MomentsEdit

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

Center of MassesEdit