Question

a. Write and simplify the integral that gives the arc length of the following curve on the given integral. b If necessary, use technology to evaluate or approximate the integral. y = sin(2x) on [0, pi/2] Set up the integral that gives the arc length of the curve. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. L = integral -1 to 1() dy B. L = integral 0 to pi/2 ()dx

Answer 1

`\displaystyle AL = \int\limits^{(\pi)/(2)}_0 {√(1+ 4cos^2(2x))} \, dx = 2.63518`

General Formulas and Concepts:

Algebra I

Functions

• Function Notation

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Integration

• Integrals
• Definite Integrals
• Integration Constant C

Integration Property [Multiplied Constant]:
`\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx`

Integration Property [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

U-Substitution

Arc Length Formula [Rectangular]:
`\displaystyle AL = \int\limits^b_a {√(1+ [f'(x)]^2)} \, dx`

Explanation:

Step 1: Define

Identify

y = sin(2x)

Interval [0, π/2]

Step 2: Find Arc Length

1. [Function] Differentiate [Trigonometric Differentiation, Chain Rule]:
   `\displaystyle (dy)/(dx) = 2cos(2x)`
2. Substitute in variables [Arc Length Formula - Rectangular]:
   `\displaystyle AL = \int\limits^{(\pi)/(2)}_0 {√(1+ [2cos(2x)]^2)} \, dx`
3. [Integrand] Simplify:
   `\displaystyle AL = \int\limits^{(\pi)/(2)}_0 {√(1+ 4cos^2(2x))} \, dx`
4. [Integral] Evaluate:
   `\displaystyle AL = 2.63518`

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

Unit: Applications of Integration

Book: College Calculus 10e