Answer:
A. 4 1/2
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Algebra I
Functions
- Function Notation
- Graphing
Solving systems of equations by graphing
Calculus
Integration
- Integrals
- Definite Integrals
- Integration Constant C
- Area under the curve
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Area of a Region Formula: [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
y = x²
y = 2 + x
Step 2: Identify
See Attachment. Find other necessary information.
Interval [-1, 2]
Step 3: Find Area
- Substitute in variables [Area of a Region Formula]: [tex]\displaystyle A = \int\limits^2_{-1} {(2 + x - x^2)} \, dx[/tex]
- [Integral] Rewrite [Integration Property - Addition/Subtraction]: [tex]\displaystyle A = \int\limits^2_{-1} {2} \, dx + \int\limits^2_{-1} {x} \, dx - \int\limits^2_{-1} x^2} \, dx[/tex]
- [1st Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle A = 2\int\limits^2_{-1} {} \, dx + \int\limits^2_{-1} {x} \, dx - \int\limits^2_{-1} x^2} \, dx[/tex]
- [Integrals] Integrate [Integration Rule - Reverse Power Rule]: [tex]\displaystyle A = 2(x) \bigg| \limits^2_{-1} + \frac{x^2}{2} \bigg| \limits^2_{-1} - \frac{x^3}{3} \bigg| \limits^2_{-1}[/tex]
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle A = 2(3) + \frac{3}{2} - 3[/tex]
- Simplify: [tex]\displaystyle A = \frac{9}{2}[/tex]
- Reduce: [tex]\displaystyle A = 4\frac{1}{2}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Book: College Calculus 10e
