Question

Which is the area between the x-axis and y=x from x=1 to x=5

Answer 1

`\displaystyle A = 12`

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

• Functions
• Function Notation
• Graphing

Calculus

Integrals

• Definite Integrals
• Area under the curve

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + C`

Integration Rule [Fundamental Theorem of Calculus 1]:
`\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)`

Area of a Region Formula:
`\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx`

Explanation:

Step 1: Define

Identify

y = x

Interval: x = 1 to x = 5

Step 2: Sort

Graph the function. See Attachment.

Bounds of Integration: [1, 5]

Step 3: Find Area

1. Substitute in variables [Area of a Region Formula]:
   `\displaystyle A = \int\limits^5_1 {x} \, dx`
2. [Integral] Integrate [Integration Rule - Reverse Power Rule]:
   `\displaystyle A = (x^2)/(2) \bigg| \limits^5_1`
3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle A = 12`

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e