Skip to main content

Integration Over the Rectangle

In the previous blog, we learned how to use single integral to calculate the volume and center of mass. For single integrals, the region over which we integrate is always an interval. (i.e. a one-dimensional space)

In this blog we are intereted in integrating a function of two variables \(f(x,y)\). With functions of two variable we integrated over a region of \(\mathbb{R}^2\) (i.e. a two-dimensional space). We will start out by assuming that the region in \(\mathbb{R}^2\) is a rectangle which we will denote as \(R=[a,b]\times [c,d]\). This means that the ranges for \(x\) and \(y\) are \(a\leq x\leq b\), and \(c \leq y\leq d\). Also let's assume \(f(x,y) \geq 0\). The graph of \(f\) is a aurface with equation \(z=f(x,y)\)
Our goal is to find the volume between surface and \(R\). First, we want to approximate it. We divide up \(R\) into a series of smaller rectangles, such that divide up \(a\leq x\leq b\) into \(n\)subintervals and divide up \(c \leq y\leq d\) into \(m\) subintervals, we denote each rectangle as \((x_i, y_j)\). Then, we will construct a box whose height is given by \(f(x_i, y_j)\) over each of these smaller rectangles. Then, each of rectangle has a base \(\Delta A\) and a height of \(f(x_i, y_j)\), so we have the volume of each boxes will be \(f(x_i, y_j) \Delta A\). The volume under \(S\) will be approximately equal to the following: \[ V \approx \sum^n_{i=1} \sum^m_{j=1} f(x_i, y_j) \Delta A\] Our intuition tells us that if the approximation becomes better as \(m\) and \(n\) become larger, we would expect that : \[ V = \lim _{m, n \to \infty} \sum^n_{i=1} \sum^m_{j=1} f(x_i, y_j) \Delta A\] It looks like the function of single integral's definition. In fact, it is the definition of double integrals. It can be written as: The double integral of \(f\) over the rectangle \(R\) is \[V= \int \int_Rf(x,y)dA = \lim _{m, n \to \infty} \sum^n_{i=1} \sum^m_{j=1} f(x_i, y_j) \Delta A\] if this limit exists. Let's use an example to learn how we use double integral to find the volume below the function and above the rectangle and how we compute double integrals.
Calculate the solid volume that lies above the square \(R=[0,2]\times[0,2]\) and below the paraboloid \(z=16-x^2-2y^2\).
First, Let's sketch this paraboloid and the square.
We want to approximate the volume of the solid. First, let's divide 100 equal squares, which means \(n=m=10\).Then we know the \(\Delta A = 0.2\times 0.2 = 0.04\) and the height of boxes will be following:
We could use 3D printer to print the model as follwoing:
Now, we know the base and height of each box, we could approximate the total volume of this solid as following: \[V \approx \sum^{10}_{i=1} \sum^{10}_{j=1} f(x_i, y_j) \Delta A = 45.52\] Let's use integral to calculate it to get an accurate result. \[\begin{split} V &=\int\int_R f(x,y) dA = \int^2_0 \int^2_0 16-x^2-2y^2 dx dy\\ &= \int^2_0 \left[16x-\frac{x^3}{3} -2xy \bigg|^{x=2}_{x=0}\right]dy = \int^2_0 32-\frac{8}{3}-4y dy \\ &= 32y-\frac{8}{3}y-4\frac{y^3}{3} \bigg |^{y=2}_{y=0} = 64-\frac{16}{3}-\frac{32}{3} \\ &= 48 \end{split} \] Comparing the approximation result with the actual result, they are closed.

I choose this function because it is an easy example for beginners to learn how to integrate over the rectangle. After they know how to integrate over the rectangle, they could continue learning how to integrate over the general regions in two dimensions.

Fig. 1 Double Integrals over Rectangles. (2020, November 10). Retrieved September 27, 2021, from https://math.libretexts.org/@go/page/4545

Author: Yueqi Li (Nicole)

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more