Post on 16-Apr-2017
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