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.

__Using Desmos to approximate a parabola to fit the top of the football:__

The function shown is blue is: $f(x)=-0.79x^2+2.8$


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

__By zooming in on the picture, it appears that the end of the football in Quadrant 1 is (5.55, 0) and since the football is symmetric about the y-axis, + and -5.55 will define the bounds for integration for the area under $f(x)$.__

__Considering $f(x)$ to be the radius of a vertical "slice" or cross-section of the football, the area of this cross section is defined as:__

$$A=2\pi(r)^2$$

$$A=2\pi(f(x))^2$$

$$A=2\pi(-0.79x^2+2.8)^2$$

__In order to find the approximate total volume of the football from [-5.55,5.55], we take the integral of the formula for the area of a single slice to sum the total areas as the number of slices approaches infinity with infinitesimally small widths ($dx$ where $\triangle(x)$ was cross section width):__

$$\int_{-5.55}^{5.55}\pi\left(-.079x^{2}+2.8\right)^{2}dx$$

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

Using Desmos to evaluate the above integral, the volume of the football shown in the picture is approximated to be 156.29 $inches^3$, which seems to be a reasonable approximation.

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To submit this assignment click on the __Publish__ button. Then copy the url of the final document and submit it in Canvas.