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Determine the area enclosed by y=2x+3, the x-axis and the ordinates x=3 and x=4​

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Answer:

[tex]\displaystyle \int\limits^4_3 {2x + 3} \, dx = 10[/tex]

General Formulas and Concepts:
Calculus

Integration

  • Integrals

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

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

Area of a Region Formula:                                                                               [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

Identify.

y = 2x + 3

x-interval [3, 4]

x-axis

See attachment for graph.

Step 2: Find Area

  1. Substitute in variables [Area of a Region Formula]:                               [tex]\displaystyle A = \int\limits^4_3 {2x + 3} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Addition/Subtraction]:           [tex]\displaystyle A = \int\limits^4_3 {2x} \, dx + \int\limits^4_3 {3} \, dx[/tex]
  3. [Integrals] Rewrite [Integration Property - Multiplied Constant]:           [tex]\displaystyle A = 2 \int\limits^4_3 {x} \, dx + 3 \int\limits^4_3 {} \, dx[/tex]
  4. [Integrals] Integrate [Integration Rule - Reverse Power Rule]:               [tex]\displaystyle A = 2 \bigg( \frac{x^2}{2} \bigg) \bigg| \limits^4_3 + 3(x) \bigg| \limits^4_3[/tex]
  5. [Integrals] Integrate [Integration Rule - FTC 1]:                                       [tex]\displaystyle A = 2 \bigg( \frac{7}{2} \bigg) + 3(1)[/tex]
  6. Simplify:                                                                                                     [tex]\displaystyle A = 10[/tex]

∴ the area bounded by the region y = 2x + 3, x-axis, and the coordinates x = 3 and x = 4 is equal to 10.

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Learn more about integration: https://brainly.com/question/26401241

Learn more about calculus: https://brainly.com/question/20197752

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

Unit: Integration

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