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.

2. Set up an integral expression that will give the volume of the football.

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

Upon looking at the image provided for a regulation size football, I proceeded to zoom into it as much as possible. I then used desmos to create a parabola with three movable points. I used the x and y-intercepts to guide. Then I used a quadratic equation with a, b, and c as sliders and matched them to the intercepts. I made two sets using the x-axis as a midline to cut the football in half. I then used the disk method and used one of the equation predictions as the radius. The bounds I used were the y-intercepts of 5.56 and -5.56. The integral expression and approximate volume is seen below:

$V=\int_{-5.55}^{5.56}\pi\left(\left(0.113423x^{2}+\left(-0.00113423x\right)+\left(-3.485\right)\right)^{2}\right)dx$

$V=225.598330595$ $in^3$


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