Question

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

Answer 1

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]:
`\displaystyle AL = \int\limits^b_a {√([x'(t)]^2 + [y(t)]^2)} \, dx`

Explanation:

Step 1: Define

Identify

`\displaystyle \left \{ {{x = t - cos(t)} \atop {y = 1 - sin(t)}} \right.`

Interval [0, π]

Step 2: Find Arc Length

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

Topic: AP Calculus BC (Calculus I + II)

Unit: Parametric Integration

Book: College Calculus 10e