Question

Write and simplify the integral that gives the arc length of the following curve on the given interval b. If necessary, use technology to evaluate or approximate the integral. y:31n x, for 2sxs5 a. The integral that gives the arc length of the curve is L dx &The arc length of the curve is approximately (Round to three decimal places as needed.)

Answer 1

a.
`\displaystyle AL = \int\limits^5_2 {\sqrt{1+ (9)/(x^2)} \, dx`

b.
`\displaystyle AL = 4.10322`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(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 = 3ln(x)

Interval [2, 5]

Step 2: Find Arc Length

1. [Function] Differentiate [Logarithmic Differentiation]:
   `\displaystyle (dy)/(dx) = (3)/(x)`
2. Substitute in variables [Arc Length Formula - Rectangular]:
   `\displaystyle AL = \int\limits^5_2 {\sqrt{1+ [(3)/(x)]^2}} \, dx`
3. [Integrand] Simplify:
   `\displaystyle AL = \int\limits^5_2 {\sqrt{1+ (9)/(x^2)} \, dx`
4. [Integral] Evaluate:
   `\displaystyle AL = 4.10322`

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

Unit: Applications of Integration

Book: College Calculus 10e