Question

Consider the following. (See attachment)

Answer 1

Answer:

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

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

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

Unit: Integration - Area under the curve

Book: College Calculus 10e