lct11 Triple integrals xyz

Simple regions R3; triple integr; mass/volume/centerofmass/centroid

What is the definition of a simple region in R3?

2 variables (for example x/y) in region D projected on xy-plane.

Limits of the third variable are dependent on the other two.

What is a type 1 region?

Projection on xy-plane.

Inner integral wrt dz.

What is a type 2 region?

Projection on yz-plane.

Inner integral wrt dx.

What is a type 3 region?

Projection on xz-plane.

Inner integral wrt dy.

What is the difference between type 1/2/3 regions?

projection on:

1. xy

2. yz

3. xz

and inner integral:

1. z

2. x

3. y

What is the mass of a volume with density rho(xyz)?

What is the volume of region E?

Give the coords of the center of mass of E with mass m and density rho(xyz).

What is the centroid of E?

center of mass with density 1.

Give formulas for moments of inertia about all 3 axes.

Note: I around 1 axis (x) → other variables in integral (x²+z²)