Answer:

General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
<u>Algebra I</u>
- Terms/Coefficients
- Expand by FOIL (First Outside Inside Last)
- Factoring
<u>Calculus</u>
Differentiation
Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integration Property:

- Fundamental Theorem of Calculus:

- Arc Length Formula:
![\displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20AL%20%3D%20%5Cint%5Climits%5Ea_b%20%7B%5Csqrt%7B1%2B%20%5Bf%27%28x%29%5D%5E2%7D%7D%20%5C%2C%20dx)
- Surface Area Formula:
![\displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20SA%20%3D%202%5Cpi%20%5Cint%5Climits%5Ea_b%20%7Bf%28x%29%20%5Csqrt%7B1%2B%20%5Bf%27%28x%29%5D%5E2%7D%7D%20%5C%2C%20dx)
Step-by-step explanation:
<u>Step 1: Define</u>
![\displaystyle y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}\\Interval: [0, \frac{1}{3}]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20%5Cfrac%7B1%7D%7B3%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20-%20x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5C%5CInterval%3A%20%5B0%2C%20%5Cfrac%7B1%7D%7B3%7D%5D)
<u>Step 2: Differentiate</u>
- Basic Power Rule:

- [Derivative] Simplify:

- [Derivative] Simplify:

<u>Step 3: Integrate Pt. 1</u>
- Substitute [Surface Area]:
![\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{1+ [\frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}]^2}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20SA%20%3D%202%5Cpi%20%5Cint%5Climits%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D_0%20%7B%28%5Cfrac%7B1%7D%7B3%7Dx%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20-%20x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%29%20%5Csqrt%7B1%2B%20%5B%5Cfrac%7B1%7D%7B6%5Csqrt%7Bx%7D%7D%20-%20%5Cfrac%7B3%5Csqrt%7Bx%7D%7D%7B2%7D%7D%5D%5E2%7D%7D%20%5C%2C%20dx)
- [Integral - √Radical] Expand/Add:

- [Integral - √Radical] Factor:

- [Integral - Simplify]:

- [Integral] Integration Property:

<u>Step 4: Integrate Pt. 2</u>
- [Integral] Define:

- [Integral] Assumption of Positive/Correction Factors:

- [Integral] Expand - FOIL:

- [Integral] Integrate - Basic Power Rule:

- [Expression] Multiply:

<u>Step 5: Integrate Pt. 3</u>
- [Integral] Substitute/Integral - FTC:

- [Integrate] Evaluate FTC:

- [Expression] Multiply:

<em>It is in ft² because it is given that our axis are in ft.</em>
<u>Step 6: Find Amount of Glass</u>
<em>Convert ft² to in² and multiply by 0.015 in (given) to find amount of glass.</em>
- Convert ft² to in²:

- Multiply:

And we have our final answer! Hope this helped on your Calc BC journey!