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Answer:

B. -15

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  

Distributive Property

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]:                                                                [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]  

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

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

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

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

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

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle \int\limits^3_{-1} {[2g(x) + 4]} \, dx = 22[/tex]

[tex]\displaystyle \int\limits^{-1}_{10} {g(x)} \, dx = 12[/tex]

[tex]\displaystyle \int\limits^{10}_{3} {g(x)} \, dx = z[/tex]

Step 2: Redefine

Manipulate the given integrals.

  1. [Integrals] Combine [Integration Property - Splitting Integral]:                    [tex]\displaystyle \int\limits^{-1}_{10} {g(x)} \, dx + \int\limits^{10}_3 {g(x)} \, dx = \int\limits^3_{10} {g(x)} \, dx[/tex]
  2. [Integrals] Rewrite:                                                                                         [tex]\displaystyle \int\limits^3_{10} {g(x)} \, dx = \int\limits^{-1}_{10} {g(x)} \, dx + \int\limits^{10}_3 {g(x)} \, dx[/tex]
  3. [Integrals] Substitute in variables:                                                                 [tex]\displaystyle \int\limits^{-1}_3 {g(x)} \, dx = 12 + z[/tex]
  4. [Integrals] Rewrite [Integration Property - Swapping Limits]:                      [tex]\displaystyle -\int\limits^3_{-1} {g(x)} \, dx = 12 + z[/tex]
  5. [Integrals] [Division Property of equality] Isolate integral:                           [tex]\displaystyle \int\limits^3_{-1} {g(x)} \, dx = -(12 + z)[/tex]
  6. [Integrals] [Distributive Property] Distribute negative:                                 [tex]\displaystyle \int\limits^3_{-1} {g(x)} \, dx = -12 - z[/tex]

Step 3: Solve

  1. [Integral] Rewrite [Integration Property - Addition]:                                     [tex]\displaystyle \int\limits^3_{-1} {2g(x)} \, dx + \int\limits^3_{-1} {4} \, dx = 22[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                   [tex]\displaystyle 2\int\limits^3_{-1} {g(x)} \, dx + 4\int\limits^3_{-1} \, dx = 22[/tex]
  3. [Integral] Substitute in integral:                                                                      [tex]\displaystyle 2(-12 - z) + 4\int\limits^3_{-1} \, dx = 22[/tex]
  4. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                      [tex]\displaystyle 2(-12 - z) + 4(x) \bigg| \limits^3_{-1} = 22[/tex]
  5. [Integral] Evaluate [Integration Rule - FTC 1]:                                                [tex]\displaystyle 2(-12 - z) + 4(3 - -1) = 22[/tex]
  6. [Integral] (Parenthesis) Simplify:                                                                    [tex]\displaystyle 2(-12 - z) + 4(3 + 1) = 22[/tex]
  7. [Integral] (Parenthesis) Add:                                                                          [tex]\displaystyle 2(-12 - z) + 4(4) = 22[/tex]
  8. [Integral] Multiply:                                                                                           [tex]\displaystyle 2(-12 - z) + 16 = 22[/tex]
  9. [Integral] [Subtraction Property of Equality] Subtract 16 on both sides:     [tex]\displaystyle 2(-12 - z) = 6[/tex]
  10. [Integral] [Division Property of Equality] Divide 2 on both sides:                [tex]\displaystyle -12 - z = 3[/tex]
  11. [Integral] [Addition Property of Equality] Isolate z term:                              [tex]\displaystyle -z = 15[/tex]
  12. [Integral] [Division Property of Equality] Isolate z:                                        [tex]\displaystyle z = -15[/tex]
  13. [Integral] Back-Substitute:                                                                              [tex]\displaystyle \int\limits^{10}_{3} {g(x)} \, dx = -15[/tex]

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

Unit: Integration

Book: College Calculus 10e

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