Section 15.1: Double Integrals over rectangles

The methods that we used for approximating single integrals (the Midpoint Rule, the Trapezoidal Rule, Simpson’s Rule) all have counterparts for double integrals. Here we consider only the Midpoint Rule for double integrals. This means that we use a double Riemann sum to approximate the double integral, where the sample point \((x_{ij}^*, y_{ij}^*)\) in \(R_{ij}\) is chosen to be the center \((\bar{x}_i, \bar{y}_j)\) of \(R_{ij}\). In other words, \(\bar{x}_i\) is the midpoint of \([x_{i-1}, x_i]\) and \(\bar{y}_j\) is the midpoint of \([y_{j-1}, y_j]\).

Midpoint Rule for Double Integrals \[ \iint_R f(x, y) dA \approx \sum_{i=1}^{m} \sum_{j=1}^{n} f(\bar{x}_i, \bar{y}_j) \Delta A \] where \(\bar{x}_i\) is the midpoint of \([x_{i-1}, x_i]\) and \(\bar{y}_j\) is the midpoint of \([y_{j-1}, y_j]\).