Question

Consider the following. (See attachment)

a) Find the area of the region
b) Use the integration capabilities of the graphing utility to verify you results. (Round your answer to one decimal place.)

Answer 1

Area: 16

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

Calculus

Derivatives

Derivative Notation

Integrals - Area under the curve

Trig Integration

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

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = 8sin(x) + sin(8x)`

`\displaystyle y = 0`

Bounds of Integration: 0 ≤ x ≤ π

Step 2: Find Area Pt. 1

1. Set up integral:
   `\displaystyle A = \int\limits^(\pi)_0 {[8sin(x) + sin(8x)]} \, dx`
2. Rewrite integral [Integration Property - Addition/Subtraction]:
   `\displaystyle A = \int\limits^(\pi)_0 {8sin(x)} \, dx + \int\limits^(\pi)_0 {sin(8x)} \, dx`
3. [1st Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle A = 8\int\limits^(\pi)_0 {sin(x)} \, dx + \int\limits^(\pi)_0 {sin(8x)} \, dx`
4. [1st Integral] Integrate [Trig Integration]:
   `\displaystyle A = 8[-cos(x)] \bigg| \limits^(\pi)_0 + \int\limits^(\pi)_0 {sin(8x)} \, dx`
5. [1st Integral] Evaluate [Integration Rule - FTC 1]:
   `\displaystyle A = 8(2) + \int\limits^(\pi)_0 {sin(8x)} \, dx`
6. Multiply:
   `\displaystyle A = 16 + \int\limits^(\pi)_0 {sin(8x)} \, dx`

Step 3: Identify Variables

Identify variables for u-substitution.

u = 8x

du = 8dx

Step 4: Find Area Pt. 2

1. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle A = 16 + (1)/(8)\int\limits^(\pi)_0 {8sin(8x)} \, dx`
2. [Integral] U-Substitution:
   `\displaystyle A = 16 + (1)/(8)\int\limits^(8\pi)_0 {sin(u)} \, du`
3. [Integral] Integrate [Trig Integration]:
   `\displaystyle A = 16 + (1)/(8)[-cos(u)] \bigg| \limits^(8\pi)_0`
4. [Integral] Evaluate [Integration Rule - FTC 1]:
   `\displaystyle A = 16 + (1)/(8)(0)`
5. Simplify:
   `\displaystyle A = 16`

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

Unit: Integration - Area under the curve

Book: College Calculus 10e