Question

Integrating sums of functions

Answer 1

(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 = (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)`

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