Triple Integral 1
is the jacobian of the transformations (x,
y, z) → (u, v, w), and S is the
pullback of G which can be computed by x = x(u, v, w), y
= y(u, v, w), z = z(u,
v, w) into the definition of G.
- Triple Integrals in Cylindrical Coodinates
The differential dxdydz for cylindrical coordinates is
Let the solid G is determined as follows:
(x, y) ∈
R, χ1 (x, y) ≤ z ≤ χ2
(x, y),
where R is projection of G onto the xy-plane. Then
Here S is the pullback of G in cylindrical coordinames.
- Triple Integrals in Spherical Coodinates
The differential dxdydz for spherical coordinates is
where the solid S is the pullback of G in spherical
coordinates. The angle θ ranges 0 to 2π, the angle φ
ranges from 0 to π.
Figure
- Volume of a Solid
- Volume in Cylindrical Coordinates
- Volume in Spherical Coordinates
- Mass of a Solid
where the solid occupies a region G and its density
at a point (x, y, z) is μ (x, y, z).
- Center of Mass of Solid
Where
are the first moments about the coordinate plane
x = 0, y = 0, z = 0, respectively, μ(x, y,z) is the density
function.
- Moments of Inertia about
the xy-plane
(or z = 0), yz-plane
(or x = 0), and xz-plane (or y= 0)
- Moments of Inertia about the x-axis, y-axis, and z-axis
- Polar Moment of Inertia
Thanks for reading Integral Calculus - Triple Integral 1. Please share...!
