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Sagot :
Answer:
B) 4√2
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- 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
- [Parametrics] Differentiate [Basic Power Rule, Trig Differentiation]: [tex]\displaystyle \left \{ {{x' = 1 + sin(t)} \atop {y' = -cos(t)}} \right.[/tex]
- Substitute in variables [Arc Length Formula - Parametric]: [tex]\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{[1 + sin(t)]^2 + [-cos(t)]^2}} \, dx[/tex]
- [Integrand] Simplify: [tex]\displaystyle AL = \int\limits^{\pi}_0 {\sqrt{2[sin(x) + 1]} \, dx[/tex]
- [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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