A school system is reducing the amount of dumpster loads of trash removed each week. In week 5, there were 40 dumpster loads of
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Answer:
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
- Left to Right
Equality Properties
- Multiplication Property of Equality
- Division Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality
<u>Algebra I</u>
- Terms/Coefficients
- Graphing
- Exponential Rule [Root Rewrite]:
<u>Calculus</u>
Derivatives
Derivative Notation
Derivative of a constant is 0
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Area - Integrals
U-Substitution
Integration Rule [Reverse Power Rule]:
Integration Property [Multiplied Constant]:
Integration Property [Addition/Subtraction]:
Integration Rule [Fundamental Theorem of Calculus 1]:
Area of a Region Formula:
Step-by-step explanation:
<u>Step 1: Define</u>
F: y = √(15 - x)
G: y = √(15 - 3x)
H: y = 0
<u>Step 2: Find Bounds of Integration</u>
<em>Solve each equation for the x-value for our bounds of integration.</em>
F
- Set <em>y</em> = 0: 0 = √(15 - x)
- [Equality Property] Square both sides: 0 = 15 - x
- [Subtraction Property of Equality] Isolate <em>x</em> term: -x = -15
- [Division Property of Equality] Isolate <em>x</em>: x = 15
G
- Set y = 0: 0 = √(15 - 3x)
- [Equality Property] Square both sides: 0 = 15 - 3x
- [Subtraction Property of Equality] Isolate <em>x</em> term: -3x = -15
- [Division Property of Equality] Isolate <em>x</em>: x = 5
This tells us that our bounds of integration for function F is from 0 to 15 and our bounds of integration for function G is 0 to 5.
We see that we need to subtract function G from function F to get our area of the region (See attachment graph for visual).
<u>Step 3: Find Area of Region</u>
<em>Integration Part 1</em>
- Rewrite Area of Region Formula [Integration Property - Subtraction]:
- [Integral] Substitute in variables and limits [Area of Region Formula]:
- [Area] [Integral] Rewrite [Exponential Rule - Root Rewrite]:
<u>Step 4: Identify Variables</u>
<em>Set variables for u-substitution for both integrals.</em>
Integral 1:
u = 15 - x
du = -dx
Integral 2:
z = 15 - 3x
dz = -3dx
<u>Step 5: Find Area of Region</u>
<em>Integration Part 2</em>
- [Area] Rewrite [Integration Property - Multiplied Constant]:
- [Area] U-Substitution:
- [Area] Reverse Power Rule:
- [Area] Evaluate [Integration Rule - FTC 1]:
- [Area] Multiply:
- [Area] Add:
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Area Under the Curve - Area of a Region (Integration)
Book: College Calculus 10e
