Question

A light bulb is designed by revolving the graph of:

Answer 1

Answer:

$\displaystyle 0.251327 \ in. \ of \ glass$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Terms/Coefficients
• Expand by FOIL (First Outside Inside Last)
• Factoring

Calculus

Differentiation

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Integration

• Integration Property: $\displaystyle \int\limits^a_b {cf(x)} \, dx = c \int\limits^a_b {f(x)} \, dx$

• Fundamental Theorem of Calculus: $\displaystyle \int\limits^a_b {f(x)} \, dx = F(b) - F(a)$

• Area between Two Curves

• Volumes of Revolution

• Arc Length Formula: $\displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx$

• Surface Area Formula: $\displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx$

Step-by-step explanation:

Step 1: Define

$\displaystyle y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}\\Interval: [0, \frac{1}{3}]$

Step 2: Differentiate

1. Basic Power Rule:                    $\displaystyle y' = \frac{1}{2} \cdot \frac{1}{3}x^{\frac{1}{2} - 1} - \frac{3}{2} \cdot x^{\frac{3}{2} - 1}$
2. [Derivative] Simplify:                $\displaystyle y' = \frac{1}{6}x^{\frac{-1}{2}} - \frac{3}{2}x^{\frac{1}{2}}$
3. [Derivative] Simplify:                $\displaystyle y' = \frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}$

Step 3: Integrate Pt. 1

1. Substitute [Surface Area]:                                                                             $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{1+ [\frac{1}{6\sqrt{x}} - \frac{3\sqrt{x}}{2}}]^2}} \, dx$
2. [Integral - √Radical] Expand/Add:                                                               $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{81x^2+18x+1}{36x}} \, dx$
3. [Integral - √Radical] Factor:                                                                         $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {(\frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}) \sqrt{\frac{(9x + 1)^2}{36x}} \, dx$
4. [Integral - Simplify]:                                                                                       $\displaystyle SA = 2\pi \int\limits^{\frac{1}{3}}_0 {-\frac{|9x + 1|(3x - 1)}{18}} \, dx$
5. [Integral] Integration Property:                                                                     $\displaystyle SA = \frac{- \pi}{9} \int\limits^{\frac{1}{3}}_0 {|9x + 1|(3x - 1)} \, dx$

Step 4: Integrate Pt. 2

1. [Integral] Define:                                                                                             $\displaystyle \int {|9x + 1|(3x - 1)} \, dx$
2. [Integral] Assumption of Positive/Correction Factors:                                 $\displaystyle \frac{9x + 1}{|9x + 1|} \int {(9x + 1)(3x - 1)} \, dx$
3. [Integral] Expand - FOIL:                                                                                 $\displaystyle \frac{9x + 1}{|9x + 1|} \int {27x^2 - 6x - 1} \, dx$
4. [Integral] Integrate - Basic Power Rule:                                                         $\displaystyle \frac{9x + 1}{|9x + 1|} (9x^3 - 3x^2 - x)$
5. [Expression] Multiply:                                                                                      $\displaystyle \frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|}$

Step 5: Integrate Pt. 3

1. [Integral] Substitute/Integral - FTC:                                                              $\displaystyle SA = \frac{- \pi}{9} (\frac{(9x + 1)(9x^3 - 3x^2 - x)}{|9x + 1|})|\limits_{0}^{\frac{1}{3}}$
2. [Integrate] Evaluate FTC:                                                                                $\displaystyle SA = \frac{- \pi}{9} (\frac{-1}{3})$
3. [Expression] Multiply:                                                                                     $\displaystyle SA = \frac{\pi}{27} \ ft^2$

It is in ft² because it is given that our axis are in ft.

Step 6: Find Amount of Glass

Convert ft² to in² and multiply by 0.015 in (given) to find amount of glass.

1. Convert ft² to in²:                    $\displaystyle \frac{\pi}{27} \ ft^2 \ \div 144 \ in^2/ft^2 = \frac{16 \pi}{3} \ in^2$
2. Multiply:                                   $\displaystyle \frac{16 \pi}{3} \ in^2 \cdot 0.015 \ in = 0.251327 \ in. \ of \ glass$

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