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.

$y=-0.099999x^{2}+3.35$

$x=±\sqrt{\frac{3.35}{0.099999}}$ or $±5.7879473912$

$$\int_{-5.7879473912}^{5.7879473912}\pi\left(0.099999x^{2}-3.35\right)^{2}dx$$

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

$V=\int_{-5.7879473912}^{5.7879473912}\pi\left(0.099999x^{2}-3.35\right)^{2}dx$

$=\pi\int_{-5.7879473912}^{5.7879473912}\left(0.099999x^{2}-3.35\right)^{2}dx$

$=\left[\pi\left(0.00199996x^{5}-0.2233311x^{3}+11.2225x\right)\right]_{-5.7879473912}^{5.7879473912}$

$= \left(\pi\left(0.00199996\left(5.7879473912\right)^{5}-0.2233311\left(5.7879473912\right)^{3}+11.2225\left(5.7879473912\right)\right)\right)-\left(\pi\left(0.00199996\left(-5.7879473912\right)^{5}-0.2233311\left(-5.7879473912\right)^{3}+11.2225\left(-5.7879473912\right)\right)\right)$

$=217.667097093$ in$^3$.

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