Answer:
![\displaystyle A = \frac{20\sqrt{15}}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cfrac%7B20%5Csqrt%7B15%7D%7D%7B3%7D)
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
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]:
![\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%5Bn%5D%7Bx%7D%20%3D%20x%5E%7B%5Cfrac%7B1%7D%7Bn%7D%7D)
<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]: ![\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bx%5En%7D%20%5C%2C%20dx%20%3D%20%5Cfrac%7Bx%5E%7Bn%20%2B%201%7D%7D%7Bn%20%2B%201%7D%20%2B%20C)
Integration Property [Multiplied Constant]: ![\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7Bcf%28x%29%7D%20%5C%2C%20dx%20%3D%20c%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx)
Integration Property [Addition/Subtraction]: ![\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%5Cpm%20%5Cint%20%7Bg%28x%29%7D%20%5C%2C%20dx)
Integration Rule [Fundamental Theorem of Calculus 1]: ![\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%5Climits%5Eb_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%3D%20F%28b%29%20-%20F%28a%29)
Area of a Region Formula: ![\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cint%5Climits%5Eb_a%20%7B%5Bf%28x%29%20-%20g%28x%29%5D%7D%20%5C%2C%20dx)
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]:
![\displaystyle A = \int\limits^b_a {f(x)} \, dx - \int\limits^d_c {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cint%5Climits%5Eb_a%20%7Bf%28x%29%7D%20%5C%2C%20dx%20-%20%5Cint%5Climits%5Ed_c%20%7Bg%28x%29%7D%20%5C%2C%20dx)
- [Integral] Substitute in variables and limits [Area of Region Formula]:
![\displaystyle A = \int\limits^{15}_0 {\sqrt{15 - x}} \, dx - \int\limits^5_0 {\sqrt{15 - 3x}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cint%5Climits%5E%7B15%7D_0%20%7B%5Csqrt%7B15%20-%20x%7D%7D%20%5C%2C%20dx%20-%20%5Cint%5Climits%5E5_0%20%7B%5Csqrt%7B15%20-%203x%7D%7D%20%5C%2C%20dx)
- [Area] [Integral] Rewrite [Exponential Rule - Root Rewrite]:
![\displaystyle A = \int\limits^{15}_0 {(15 - x)^{\frac{1}{2}}} \, dx - \int\limits^5_0 {(15 - 3x)^{\frac{1}{2}}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cint%5Climits%5E%7B15%7D_0%20%7B%2815%20-%20x%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20-%20%5Cint%5Climits%5E5_0%20%7B%2815%20-%203x%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5C%2C%20dx)
<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]:
![\displaystyle A = -\int\limits^{15}_0 {-(15 - x)^{\frac{1}{2}}} \, dx + \frac{1}{3}\int\limits^5_0 {-3(15 - 3x)^{\frac{1}{2}}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20-%5Cint%5Climits%5E%7B15%7D_0%20%7B-%2815%20-%20x%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5C%2C%20dx%20%2B%20%5Cfrac%7B1%7D%7B3%7D%5Cint%5Climits%5E5_0%20%7B-3%2815%20-%203x%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5C%2C%20dx)
- [Area] U-Substitution:
![\displaystyle A = -\int\limits^0_{15} {u^{\frac{1}{2}}} \, du + \frac{1}{3}\int\limits^0_{15} {z^{\frac{1}{2}}} \, dz](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20-%5Cint%5Climits%5E0_%7B15%7D%20%7Bu%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5C%2C%20du%20%2B%20%5Cfrac%7B1%7D%7B3%7D%5Cint%5Climits%5E0_%7B15%7D%20%7Bz%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%7D%20%5C%2C%20dz)
- [Area] Reverse Power Rule:
![\displaystyle A = -(\frac{2u^{\frac{3}{2}}}{3}) \bigg|\limit^0_{15} + \frac{1}{3}(\frac{2z^{\frac{3}{2}}}{3}) \bigg|\limit^0_{15}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20-%28%5Cfrac%7B2u%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B3%7D%29%20%5Cbigg%7C%5Climit%5E0_%7B15%7D%20%2B%20%5Cfrac%7B1%7D%7B3%7D%28%5Cfrac%7B2z%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7D%7B3%7D%29%20%5Cbigg%7C%5Climit%5E0_%7B15%7D)
- [Area] Evaluate [Integration Rule - FTC 1]:
![\displaystyle A = -(-10\sqrt{15}) + \frac{1}{3}(-10\sqrt{15})](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20-%28-10%5Csqrt%7B15%7D%29%20%2B%20%5Cfrac%7B1%7D%7B3%7D%28-10%5Csqrt%7B15%7D%29)
- [Area] Multiply:
![\displaystyle A = 10\sqrt{15} + \frac{-10\sqrt{15}}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%2010%5Csqrt%7B15%7D%20%2B%20%5Cfrac%7B-10%5Csqrt%7B15%7D%7D%7B3%7D)
- [Area] Add:
![\displaystyle A = \frac{20\sqrt{15}}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%20%3D%20%5Cfrac%7B20%5Csqrt%7B15%7D%7D%7B3%7D)
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Area Under the Curve - Area of a Region (Integration)
Book: College Calculus 10e