Math 182 Miniproject 4 The Volume of a Football.md
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Math 182 Miniproject 4 The Volume of a Football
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**Overview:** In this project we find exact formulas for integral approximations using Riemann sums of various flavors.

**Prerequisites:** Section 6.2 of _Active Calculus_.

Go to https://www.desmos.com/calculator/c7aip1g33m to see a regulation size football image. (All units are inches.)

1. Using your graph plotting kung fu, find a curve that approximates the boundary of (at least part of) the football.

I estimated a few points starting from the furthest left section of the football, and I fit a curve of the square root function to it using Desmos.

The function is
$$f(x)=0.157498\sqrt{89.4118x+496.057}-0.0206634$$

I will find the volume of the football by slicing it into vertical slices from -5.548 to 0. Then, I will multiply the volume by two to get the other half of the football.

2. Set up an integral expression that will give the volume of the football.
$$Area = \pi r^2$$
$$=\pi (0.157498\sqrt{89.4118x+496.057}-0.0206634)^2$$
$$Volume = \pi \int_{-5.548}^{0} (0.157498\sqrt{89.4118x+496.057}-0.0206634)^2 dx$$

3. Use Desmos to find the value of your integral. What is the volume of the football?

The volume of half of the football is $105.558382047in^3$
so we will multiply the number by two for the total volume of the football which is $211.116764094in^3$

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