Answered

Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Please help with this question and give an explanation if possible

Please Help With This Question And Give An Explanation If Possible class=

Sagot :

Space

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

We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.