DOUBLE AND TRIPLE INTEGRALSonendra Gupta

Associate ProfessorDepartment of mathematics

Author of 17 Books of MathematicsVisit:sonendragupta.blogspot.in

introduction

Example

1.) Find the moment of inertia about the z-axis of the solid that lies below the paraboloid         z  =  25 – x2 - y2 inside the cylinder 

        x2 + y2  =  4above the xy-plane, and has density function        (x,y,z)  =  x2 + y2 + 6z           

Solution    By the moment of inertia formula, we have       

The region, being inside of a cylinder is ripe for cylindrical coordinates.  We get        

ExampleFind the volume of solid that lies inside the sphere 

        x2 + y2 + z2  =  2and outside of the cone

        z2  =  x2 + y2 

SolutionWe convert to spherical coordinates.  The sphere becomes 

          = 

To convert the cone, we add z2 to both sides of the equation        2z2  =  x2 + y2 +z2 Now convert to        22cos  =  2 Canceling the 2 and solving for we get          =  cos-1(1/ )  =  /4  or 7/4  

In spherical coordinates (since the coordinates are periodic)        7/4  =  3/4

To find the volume we compute               

32 24

2

0 04

sin V d d d

 

Evaluating this integral should be routine at this point and is equal to                   8     V  =     —                             3