Triple Riemann Sums

Our next step is to define the triple Riemann sum. Given positive real numbers Δx, Δy, and Δz, we partition the circumscribed rectangular box of E into rectangular boxes with sides Δx, Δy, and Δz (Figure 12.6.3). The partition points of this three-dimensional partition have the form

(x_k,y_l,z_m) 0 ≤ k ≤ n, 0 ≤ l ≤p, 0 ≤ m ≤ q.

The triple Riemann sum of f(x, y, z) Δx Δy Δz over E is defined as the sum

When we replace Δx, Δy, Δz by positive infinitesimals dx, dy, dz we obtain an infinite triple Riemann sum

Figure 12.6.3

Figure 12.6.4

INDEPENDENCE OF dx, dy, AND dz

The value of does not depend on dx, dy, or dz.

We shall usually use the notation dV = dxdydz for the volume of an infinitesimal dx by dy by dz rectangular box. and write

for

ADDITION PROPERTY

If E is divided into two regions E₁ and E₂ which meet only on a common boundary then