HUGE Waves

When starting to think about 2D integration over a specific region in the plane, I thought about indoor surfing. I knew I wanted to choose a function that increased exponentially because I thought it would give us something better to look at, so I imagined waves in the ocean. Since waves in the ocean are not restricted to a specific region and they continue on forever, I had to come up with a way to modify my example to show the restraints. This was my thought process when thinking of indoor surfing. Although I've never been indoor surfing myself, I like the idea of it. You are surfing in a controlled wave that is in a controlled region or pool area. The photo below shows an example of an indoor surfing arena.

From this picture you can see that most indoor surfing places put the wave in more of a square or rectangular region, but I wanted to spruce up my indoor surfing region and make it a quarter circle. When thinking about 2D integration over a region, it helps to think about an indoor surfing arena. The water or wave outline make up the function, the pool that the wave is restricted to is the region in the plane, and the water filling the pool and up to the top of the wave represents the volume making up the entire object. The following f(x) is the function that makes up my wave and D is the domain in which the wave is restricted: \begin{aligned}f\left( x\right) =ye^{x-2}\\ D=\left\{ \left( x,y\right) | x^{2}+y^{2}\leq 5^{2},x\geq 0,y\geq 0\right\} \end{aligned} I knew I wanted to choose a function that goes from a very subtle increase to a major increase to mimic that of a good surfing wave, so I chose a function with an exponential increase. I chose the domain to spruce up the basic wave pool shape.

We can solve for the approximate volume of our shape by breaking it up into multiple rectangular prisms and adding the volume of each prism together. I chose to use the upper left hand corner of each of the rectangular prisms square bases in order to include as many prisms as possible. Because of this, I expect my estimated volume to be greater than the actual volume of the shape. Below is a table showing the volumes for each rectangular prism and a total estimated volume of 258.56 units cubed:

The actual volume of the object can be calculated by using a double integral and integrating with respect to y and then x. Below is the math used to calculate the actual volume to be 270.69 units cubed: \begin{aligned}\int ^{6}_{0}\int _{0}^{\sqrt{36-x^{2}}}ye^{x-2}dydx\\ =\int _{0}^{6}\dfrac{1}{2}y^{2}e^{x-2}| _{0}^{\sqrt{36-x^{2}}}dx\\ =\int _{0}^{6}\dfrac{1}{2}\left( \sqrt{36-x^{2}}\right) ^{2}e^{x-2}dx\\ =\int _{0}^{6}\dfrac{1}{2}\left( 36-x^{2}\right) e^{x-2}dx\\ =\dfrac{10e^{6}-34}{2e^{2}}=270.69units^{3}\end{aligned} My actual volume actually happens to be less than the estimated which means if I were to have chosen the upper right hand points, I would have gotten an over estimate. After creating my object on Onshape, I realized that my wave increases a little too rapidly and it will be too tall to print, so before I print, I will have to scale down my object to make it a little shorter. It's current dimensions are 6 x 6 x 60 inches. The following picture is what my printed object would look like if I do not scale it down. The height of the wave increases as x increases and as y increases.