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What is the length of the curve with parametric equations x = t - cos(t), y = 1 - sin(t) from t = 0 to t = π? (5 points)
A) 8
B) 4 times the square root of 2
C) 4
D) 8 times the square root of 2

Sagot :

Spaceidn

Answer:

B) 4√2

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Parametric Differentiation

Integration

  • Integrals
  • Definite Integrals
  • Integration Constant C

Arc Length Formula [Parametric]:                                                                         [tex]\displaystyle AL = \int\limits^b_a {\sqrt{[x'(t)]^2 + [y(t)]^2}} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \left \{ {{x = t - cos(t)} \atop {y = 1 - sin(t)}} \right.[/tex]

Interval [0, π]

Step 2: Find Arc Length

  1. [Parametrics] Differentiate [Basic Power Rule, Trig Differentiation]:         [tex]\displaystyle \left \{ {{x' = 1 + sin(t)} \atop {y' = -cos(t)}} \right.[/tex]
  2. Substitute in variables [Arc Length Formula - Parametric]:                       [tex]\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{[1 + sin(t)]^2 + [-cos(t)]^2}} \, dx[/tex]
  3. [Integrand] Simplify:                                                                                       [tex]\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{2[sin(x) + 1]} \, dx[/tex]
  4. [Integral] Evaluate:                                                                                         [tex]\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{2[sin(x) + 1]} \, dx = 4\sqrt{2}[/tex]

Topic: AP Calculus BC (Calculus I + II)

Unit: Parametric Integration

Book: College Calculus 10e

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