Double Riemann Sum - Generalized

Now let D be a general region

a₁ ≤ x ≤ a₂, b₁(x) ≤ y ≤ b₂(x).

The circumscribed rectangle of D is the rectangle

a₁ ≤ x ≤ a₂, B₁ ≤ y ≤ B₂,

where

B₁ = minimum value of b₁(x),

B₂ = maximum value of b₂(x).

It is shown in Figure 12.1.8.

Figure 12.1.8 The circumscribed rectangle

Given positive real numbers Δx and Δy we partition the circumscribed rectangle of D into Δx by Δy subrectangles with partition points

(x_k, y_l), 0 ≤ k ≤ n, 0 ≤ l ≤ p.

Notice that in the double Riemann sum over D, we only use partition points (x_k, y_l) which belong to D (Figure 12.1.9).

Figure 12.1.9 Double Riemann Sum

Example 1

Example 2