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.