Skip to main content

Integration for Over Regions in the Plane

    Integrating functions of two variables can be difficult to understand at first because we are integrating in the x and y planes. By doing this, it allows for us to calculate the volume under the function in a 3D space, compared to the area of a 2D space, which would be done through a single integral. To solve a double integral, first integrate the function in one region along the bounds (while pretending the other variable is a constant), then integrate on the other region using the bounds for that region.

For my example, let’s use the function

bounded on  

 The domain (the bounds) tells us to integrate this function from -3 to 3 in the x direction and -1 to 3 in y direction. This allows us to integrate in the shape of a rectangle focusing on 1x1 blocks for each point. The function F(x,y) will calculate the height for each part of the function. This will form an object that represents the volume of F(x,y).

Approximate Volume:

To calculate the approximate volume and understand a visual representation of this function, we will use every coordinate in [-3,3] x [-1,3]. Using this domain, we will calculate the height at each coordinate and apply it to the 1x1 block with the matching coordinate to find the approximate volume under F(x,y). To do so, we plug in the x and y coordinates into the function to get the approximate individual volumes. To find the total approximate volume, we sum the individual volumes.  


The total approximate volume is 5.435 in^3. 

Applying these values to our 3D model, the approximate height calculated is used to on the bottom left corner of each block. There will be 35 blocks to build the model.

Actual Volume:

To calculate the actual volume, we are going to solve the following double integral. 


 

We can integrate in terms or x or y first (doesn’t matter), but today I’m going to start with in terms of y.

   

  


The actual volume is 1.106 in^3

Something to take note of is the difference between the actual and approximate volumes. There will be a larger difference because we are approximating using the bottom left corner of each block.

Why I Chose This Function:

I chose this function because I wanted to create a model that was visually interesting. Integrating the cosine function among two variables resulted in lots of vary in height that created a curve or wave shape. It was interesting to see how the x and y coordinates both affected the trig function and create some sort of pattern that is constantly changing. This was very cool to see compared to other models that appear more linear. Here is a visual of a smooth curve from Wolfram Alpha and my model in Onshape.



I decided to keep my domain a rectangle because I wanted to see the different changes that occur when x and y are negative too. If the domain would have occurred on another function, it might limit this representation.

At first, I thought it would be challenging to integrate the function, but it was simple trig integrals. Which was also another reason for this choice because I haven’t used a trig function this semester and wanted to try it out. Overall, I wanted a function that would produce an interesting visual, but also still be a good example for integrating on a double integral.


Comments

Popular posts from this blog

The Approximation of a Solid of Revolution

Most math teachers I've had have been able to break down Calculus into two very broad categories: derivatives and integrals. What is truly amazing, is how much you can do with these two tools. By using integration, it is possible to approximate the shape of a 2-D function that is rotated around an axis. This solid created from the rotation is known as a solid of revolution. To explain this concept, we will take a look at the region bounded by the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] bounded at the line y = 1. This region is meant to represent a cross section of a small bowl. While it may not perfectly represent this practical object, the approximation will be quite textured, and will provide insight into how the process works. The region bounded by the two functions can be rotated around the y-axis to create a fully solid object. This is easy enough to talk about, but what exactly does this new solid look like? Is...

Solids of Revolution Revisited

Introduction In my previous blog post on solids of revolution, we looked at the object formed by rotating the area between \(f(x) = -\frac{1}{9}x^2+\frac{3}{4} \) and \( g(x)=\frac{1}{2}-\frac{1}{2}e^{-x} \) around the \( x \) axis and bounded by \( x = 0 \) and \( x = 1.5 \). When this solid is approximated using 10 washers, the resulting object looks like this: When I was looking back over the 3D prints I’d created for this course, I noticed that the print for this example was the least interesting of the bunch. Looking at the print now, I feel like the shape is rather uninteresting. The curve I chose has such a gradual slope that each of the washers are fairly similar in size and causes the overall shape to just look like a cylinder. Since calculating the changes in the radiuses of the washers is a big part of the washer method, I don’t think this slowly decreasing curve was the best choice to illustrate the concept. The reason I had done this o...

Finding the Center of Mass of a Toy Boat

Consider two people who visit the gym a substantial amount. One is a girl who loves to lift weights and bench press as much as she possibly can. The other is a guy who focuses much more on his legs, trying to break the world record for squat weight. It just so happens that these two are the same height and have the exact same weight, but the center of their weight is not in the same part of their body. This is because the girl has much more weight in the top half of her body and the boy has more weight in the bottom half. This difference in center of mass is a direct result of the different distributions of mass throughout both of their bodies. Moments and Mass There are two main components to finding the center of mass of an object. The first, unsurprisingly, is the mass of the whole object. In this case of the boat example, the mass will be uniform throughout the entire object. This is ideal a majority of the time as it drastically reduces the difficulty...