How to use the midpoint rule for double integrals

The midpoint rule for a single integral uses rectangles to estimate area. For a double integral, the rectangles become bases of boxes. We’ll use the function value at each base’s midpoint to choose the height of its box.

Divide the region and find the area of each base

On $R = [a, b] \times [c, d]$ , divide the x-interval into $m$ equal parts and the y-interval into $n$ equal parts. The widths are $Δ x = (b - a) / m$ and $Δ y = (d - c) / n$ . Each base has area $Δ A = Δ x Δ y$ .

Let $x_{i}^{*}$ and $y_{j}^{*}$ be the midpoints of those intervals. Use $f (x_{i}^{*}, y_{j}^{*})$ as the height for that base. When $f$ is nonnegative, the sum of the box volumes approximates the volume under the surface.

\begin{array}{l} \iint_{R} f (x, y) d A \\ \approx \sum_{i = 1}^{m} \sum_{j = 1}^{n} f (x_{i}^{*}, y_{j}^{*}) Δ A \end{array}

Find the four midpoints in the example

Let $f (x, y) = x + y^{2} + 2$ on $R = [0, 2] \times [0, 4]$ with $m = n = 2$ . Split at $x = 1$ and $y = 2$ . The x-midpoints are $1 / 2$ and $3 / 2$ . The y-midpoints are $1$ and $3$ . Each rectangle has $Δ A = (1) (2) = 2$ .

Each rectangle is 1 unit wide and 2 units tall. The dots mark the four sample points.

Evaluate the height at each midpoint

At midpoint A, for example, $f (1 / 2, 1) = 1 / 2 + 1^{2} + 2 = 7 / 2$ . Do the same for the other three points.

Heights above the four midpoints
Point	Midpoint	Height
A	$(\frac{1}{2}, 1)$	$\frac{7}{2}$
B	$(\frac{3}{2}, 1)$	$\frac{9}{2}$
C	$(\frac{1}{2}, 3)$	$\frac{23}{2}$
D	$(\frac{3}{2}, 3)$	$\frac{25}{2}$

The four bases have the same area, so we can add the heights first and multiply by $2$ .

2 (\frac{7}{2} + \frac{9}{2} + \frac{23}{2} + \frac{25}{2}) = 64

The estimated volume is $64$ cubic units. For comparison, the exact volume is $200 / 3$ cubic units. You get that by integrating in $x$ from $0$ to $2$ , then in $y$ from $0$ to $4$ .

How to use the midpoint rule for double integrals

Divide the region and find the area of each base

Find the four midpoints in the example

Evaluate the height at each midpoint

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