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

Alternative way:

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

gives

or