A light bulb is designed by revolving the graph of:

[tex]y = \frac{1}{3}x^{\frac{1}{2}} - x^{\frac{3}{2}}[/tex]

Over the interval 0 ≤ x ≤ 1/3 about the x-axis, where x and y are measured in feet. Find the surface area of the bulb and use the result to approximate the amount of glass needed to make the bulb. (Glass is 0.015 inch thick).

Answers

Answer 1

Answer:

[tex]\displaystyle 0.251327 \ in. \ of \ glass[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Algebra I

Terms/CoefficientsExpand by FOIL (First Outside Inside Last)Factoring

Calculus

Differentiation

Derivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integration Property: [tex]\displaystyle \int\limits^a_b {cf(x)} \, dx = c \int\limits^a_b {f(x)} \, dx[/tex]Fundamental Theorem of Calculus: [tex]\displaystyle \int\limits^a_b {f(x)} \, dx = F(b) - F(a)[/tex]Area between Two CurvesVolumes of RevolutionArc Length Formula: [tex]\displaystyle AL = \int\limits^a_b {\sqrt{1+ [f'(x)]^2}} \, dx[/tex]Surface Area Formula: [tex]\displaystyle SA = 2\pi \int\limits^a_b {f(x) \sqrt{1+ [f'(x)]^2}} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

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

Step 2: Differentiate

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

Step 3: Integrate Pt. 1

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

Step 4: Integrate Pt. 2

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

Step 5: Integrate Pt. 3

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

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.

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

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


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Answer:

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Step-by-step explanation:

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Answer:

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Answer:

Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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Answers

Answer:

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Answers

Answer:

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Step-by-step explanation:

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Answers

Answer:

The slope is 3.

Step-by-step explanation:

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Answer:

Slope =

6

2

Step-by-step explanation:

First find the rise over run

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———-

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——

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Next solve

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Answers

Answer:

x < 5

Step-by-step explanation:

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Answer:

any number under 5

Step-by-step explanation:

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Is this what you're asking?

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Answers

Answer:

b -3,3 c 6,9 hope this helps

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Answers

Answer:

30

Step-by-step explanation:

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Answer:

A  76

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7x+8+7x-2+5x+16+110=360

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Answer:

76. hope it helps.................

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Step-by-step explanation:

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Line 1: Undefined

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Line 3: 4/3

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Line 8: 0

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Step-by-step explanation:

Use the slope formula (you can just search up the slope formula)

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Answer:

7 ÷ 84 = 84 = 12 ounces

             _

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Step-by-step explanation:

The equation will be 7x = 84. Then the number of ounces will each person get to drink will be 12.

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Answers

Answer:

-2

Step-by-step explanation:

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Answers

Answer:

[tex]37.7\ \text{foot}[/tex]

Step-by-step explanation:

The length of the edging will be equal to the circumference of the circular flower garden.

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Answers

Answer:

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Answer:

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Step-by-step explanation:

Answer:

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A number base is the number of digits or combination of digits that a system of counting uses to represent numbers. A base can be any whole number greater than 0. The most commonly used number system is the decimal system, commonly known as base 10. ... Binary is the most commonly used non-base 10 system.

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