Question

Integrating sums of functions

Answer 1

Answer:

(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

Equality Properties

• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality
• Subtraction Property of Equality

Calculus

Integrals

Integration Rule [Reverse Power Rule]:                                                                    $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Swapping Limits]:                                                                $\displaystyle \int\limits^b_a {f(x)} \, dx = -\int\limits^a_b {f(x)} \, dx$

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

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

Integration Property [Splitting Integral]:                                                                $\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx$

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

Step-by-step explanation:

Step 1: Define

$\displaystyle \int\limits^6_4 {f(x)} \, dx = 5$

$\displaystyle \int\limits^4_{10} {f(x)} \, dx = 8$

$\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx$

Step 2: Solve Pt. 1

1. [Integral] Rewrite [Integration Property - Addition]:                                     $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = \int\limits^{10}_6 {4f(x)} \, dx + \int\limits^{10}_6 {10} \, dx$
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                   $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4\int\limits^{10}_6 {f(x)} \, dx + 10\int\limits^{10}_6 {} \, dx$

Step 3: Redefine

Manipulate the given integral values.

1. [Integrals] Combine [Integration Property - Splitting Integral]:                     $\displaystyle \int\limits^6_4 {f(x)} \, dx + \int\limits^4_{10} {f(x)} \, dx = \int\limits^6_{10} {f(x)} \, dx$
2. [Integral] Rewrite:                                                                                           $\displaystyle \int\limits^6_{10} {f(x)} \, dx = \int\limits^6_4 {f(x)} \, dx + \int\limits^4_{10} {f(x)} \, dx$
3. [Integral] Substitute in integrals:                                                                    $\displaystyle \int\limits^6_{10} {f(x)} \, dx = 5 + 8$
4. [Integral] Add:                                                                                                 $\displaystyle \int\limits^6_{10} {f(x)} \, dx = 13$
5. [Integral] Rewrite [Integration Property - Swapping Limits]:                        $\displaystyle -\int\limits^{10}_6 {f(x)} \, dx = 13$
6. [Integral] [Division Property of Equality] Isolate integral:                             $\displaystyle \int\limits^{10}_6 {f(x)} \, dx = -13$

Step 4: Solve Pt. 2

1. [Integral] Substitute in integral:                                                                     $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10\int\limits^{10}_6 {} \, dx$
2. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                      $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10(x) \bigg| \limits^{10}_6$
3. [Integral] Evaluate [Integration Rule - FTC 1]:                                               $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10(10 - 6)$
4. [Integral] (Parenthesis) Subtract:                                                                   $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10(4)$
5. [Integral] Multiply:                                                                                           $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = -52 + 40$
6. [Integral] Add:                                                                                                 $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = -12$

Topic: AP Calculus AB/BC

Unit: Integration

Book: College Calculus 10e