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I really need this answr tonight please HELP!!! Use limits to find the area between the graph of the function and the x axis given by the definite integral. ∫_1^5▒(x^2-x+1)dx

Respuesta :

Space

Answer:

[tex]\displaystyle \int\limits^5_1 {(x^2 - x + 1)} \, dx = \frac{100}{3}[/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 [Addition/Subtraction]:                                                       [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int\limits^5_1 {(x^2 - x + 1)} \, dx[/tex]

Step 2: Integrate

  1. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               [tex]\displaystyle \int\limits^5_1 {(x^2 - x + 1)} \, dx = \int\limits^5_1 {x^2} \, dx - \int\limits^5_1 {x} \, dx + \int\limits^5_1 {} \, dx[/tex]
  2. [Integrals] Integration Rule [Reverse Power Rule]:                                     [tex]\displaystyle \int\limits^5_1 {(x^2 - x + 1)} \, dx = \bigg( \frac{x^3}{3} - \frac{x^2}{2} + x \bigg) \bigg| \limits^5_1[/tex]
  3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           [tex]\displaystyle \int\limits^5_1 {(x^2 - x + 1)} \, dx = \frac{100}{3}[/tex]

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

Unit: Integration

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