Unit 5: Double Integral Over Rectangular Regions

The Concept

The definition of the single integral in 2D space is as follows: given a single-variable function [latex]y=f(x)[/latex] that is continuous on the interval [latex][a,b][/latex], we divide the interval into [latex]n[/latex] subintervals of equal width, [latex]x[/latex], and from each interval choose a point, [latex]x_i[/latex]. Definite integral, [latex]\int_a^bf(x)dx[/latex], represents the area inbetween the curve, [latex]y=f(x)[/latex], and [latex]x[/latex]-axis. Riemann sum helps to approximate such areas, that is,

[latex]\int_a^bf(x)dx \approx \sum_{i=1}^n f(x_i) \Delta x[/latex],
where [latex]\Delta x = \frac{b-a}{n}[/latex] and [latex]x_i=a + i\Delta x[/latex]. The larger [latex]n[/latex] is, the better the estimation is. Thus, the limit of the Riemann sum defines the definite integral,

[latex]\int_a^bf(x)dx =\lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x = \sum_{i=1}^{\infty} f(x_i)\,\Delta x[/latex].

Similar to the single integral, the double integral in 3D, [latex]\iint_R f(x,y) dA[/latex], is equal to the volume under the surface of the two-variable function [latex]z = f(x,y)[/latex] and above the region [latex]R[/latex] on the [latex]xy[/latex]-plane. Here, we consider this region has a very simple shape, rectangle, and use [latex]R[/latex] to denote it. The [latex]x[/latex] coordinate of this rectangle changes from [latex]a[/latex] to [latex]b[/latex], and [latex]y[/latex] coordinate changes from [latex]c[/latex] to [latex]d[/latex], denoted as [latex]R=[a, b]\times[c,d][/latex]. As in the case of the single integral, a double integral is defined as the limit of a Reimann sum, i.e.,

[latex]\iint_R f(x,y) dA=\lim_{m,n \to \infty}\sum_{i=1}^{m} \sum_{j=1}^{n} f(x_{i},y_{j}) \Delta A[/latex]

where [latex]\Delta A=\Delta x \Delta y, \Delta x=\frac{b-a}{n}, \Delta y=\frac{d-c}{m}, x_i=a + i\Delta x[/latex] and [latex]y_i=c + i\Delta y[/latex].

The Plot

Now, you should engage with the 3D plot below to understand double integrals over rectangular regions^[1]. Follow the steps below to apply changes to the plot and observe the effects:

1. Input a function of two variables into the [latex]f(x,y)[/latex] input function section.
2. Move the [latex]n[/latex]-slide around to decide the subregions of the rectangular region, [latex]R[/latex], and we consider the subregions are squares.
3. Pick xmin, xmax, ymin, and ymax points for your domain/bounds of the rectangular region, [latex]R[/latex].
4. Use the [latex]k[/latex]-slider to choose which square-shaped subregion you’d like to highlight.
5. Use the checkboxes to show either all of the rectangular prisms compared to just the one you are highlighting, as well as whether to see the graph or not.
6. By changing your view and hovering over the plot, you can see a 2D representation of the rectangular area. Additionally, the double integral is dynamically calculated at the bottom.