Lecture 22, Monday, 10/20/08: Section 13.5: Triple integrals in rectangular coordinates

Topics

• Triple integrals in rectangular coordinates
• Volumes by triple integrals
• Applications: Mass, center of mass, and moments of inertia (3D case)

Read

Homework

Notes

• General remarks on triple integrals: Sections 13.5 - 13.7 cover triple integrals in various coordinate systems (rectangular, cylindrical, and spherical). These sections are among the most important, but also the most difficult, sections in the book. The definition and computation of triple integrals is largely analogous to that of double integrals, as are most of the formulas and applications (e.g., volume, mass, center of mass, moment of inertial).

  As with double integrals, the key (and conceptually most difficult part) is the correct set-up of a triple integral as an iterated integral, with appropratiate limits on each of the integration signs. With triple integrals this is more difficult since the regions of integration are 3-dimensional and therefore harder to sketch. (It is usually best to just sketch the 2-dimensional region R in the xy-plane over which the solid lies, and perhaps also a vertical slice, plotting z against x or y or r, instead of the full 3-dimensional solid.)

  An additional difficulty in triple integrals can be the evaluation of the iterated integrals, which is often tedious and lengthy (though conceptually not hard). Therefore, in many of the problems I will only ask for the set-up of a triple integral as an iterated integral, and not the complete evaluation of the integral.

• Mass, center of mass, and moment of inertia. These are important applications in mechanics that come up, for example, in classes like TAM 210/211.

  The formulas for mass and center of mass of a 3-dimensional solid (given a mass density function rho(x,y,z)) are completely analogous to those in the 2-dimensional case. The same goes for the moment of inertia with respect to a given axis if one keeps in mind that the integrand is the square of the distance to the axis. For example, in 3 dimensions, the moment of inertia I_x with respect to the x-axis is the triple integral over (y² + z²) rho (x,y,z). (In 2 dimensions, there would be no z² term.) Similarly for I_y and I_z, the moments of inertia with respect to the y- and z-axes. The moment of inertia with respect to the z-axis (with integrand (x² + y²) rho (x,y,z).) is also called the polar moment of inertia.

  For inexplicable reasons, the book treats mass and center of mass in the 3-dimensional case, but not moments of inertia. You are expected to know the latter as well. Since, as mentioned, the formulas are analogous to those in the 2-dimensional case, this is not a big deal.

Back to the Course Homepage

Last modified Tue 21 Oct 2008 11:41:30 AM CDT