Triple Integrals in Rectangular CoordinatesTriple Integrals in Rectangular Coordinates
Similar to double integration
If E is of type I region given by equation 5, then
Triple Integrals in Cylindrical
Coordinates
Note 1:
Note 2:
Note 3:
Note 4:
Suppose that E is a type 1 region whose
projection D onto the xy-plane is conveniently
described in polar coordinates.
Evaluating Triple Integrals with Cylindrical Coordinates
In particular, suppose that f is continuous and
E = {(x, y, z)|(x, y) ∈ D, u1(x, y) ≤≤≤≤ z ≤≤≤≤ u2(x, y)}
where D is given in polar coordinates by
D = {(r, θ)|α ≤≤≤≤ θ ≤≤≤≤ β , h (θ) ≤≤≤≤ r ≤≤≤≤ h (θ)}D = {(r, θ)|α ≤≤≤≤ θ ≤≤≤≤ β , h1(θ) ≤≤≤≤ r ≤≤≤≤ h2(θ)}
We know
But to evaluate double integrals in polar
coordinates, we have the formula
Triple Integrals in Spherical
Coordinates
Note 1:
Note 2:
Note 3:
Substitutions in Multiple Integrals