9-24 Find the exact length of the curve. 9. y = x³/2, 0≤x≤2 10. y = (x + 4) ³/2, 11. y = (1 + x²) ³/², 12. 36y² = (x² − 4)³, 13. y = 14. x = 3 talo 14 8 + + - 4x' 1 4y²² 0≤x≤ 4 0≤x≤ 1 2≤x≤ 3, 2 ≤ x ≤ 3, y ≥0 1≤x≤2 1≤ y ≤2 15. yIn(sin 2x), π/8 ≤ x ≤ π/6 16. y = ln(cos x), 0≤x≤ π/3
9-24 Find the exact length of the curve. 9. y = x³/2, 0≤x≤2 10. y = (x + 4) ³/2, 11. y = (1 + x²) ³/², 12. 36y² = (x² − 4)³, 13. y = 14. x = 3 talo 14 8 + + - 4x' 1 4y²² 0≤x≤ 4 0≤x≤ 1 2≤x≤ 3, 2 ≤ x ≤ 3, y ≥0 1≤x≤2 1≤ y ≤2 15. yIn(sin 2x), π/8 ≤ x ≤ π/6 16. y = ln(cos x), 0≤x≤ π/3
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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9,13
![**Section 9–24: Curve Length Calculation**
In this section, we'll find the exact length of each given curve. The curves specified are provided along with their respective intervals of \( x \) or \( y \). Each function and its interval are listed below:
1. **Curve 9**:
\[ y = \frac{2}{3}x^{3/2}, \quad 0 \leq x \leq 2 \]
2. **Curve 10**:
\[ y = (x + 4)^{3/2}, \quad 0 \leq x \leq 4 \]
3. **Curve 11**:
\[ y = \frac{2}{3}(1 + x^2)^{3/2}, \quad 0 \leq x \leq 1 \]
4. **Curve 12**:
\[ 36y^2 = (x^2 - 4)^3, \quad 2 \leq x \leq 3, \quad y \geq 0 \]
5. **Curve 13**:
\[ y = \frac{x^3}{3} + \frac{1}{4x}, \quad 1 \leq x \leq 2 \]
6. **Curve 14**:
\[ x = \frac{y^4}{8} + \frac{1}{4y^2}, \quad 1 \leq y \leq 2 \]
7. **Curve 15**:
\[ y = \frac{1}{2} \ln(\sin 2x), \quad \frac{\pi}{8} \leq x \leq \frac{\pi}{6} \]
8. **Curve 16**:
\[ y = \ln(\cos x), \quad 0 \leq x \leq \frac{\pi}{3} \]
### Explanation of Tasks:
1. **Integral Form for Arc Length**:
The exact length \( L \) of a curve given by \( y = f(x) \) over an interval \( [a, b] \) is calculated using the formula:
\[
L = \int_{a}^{b} \sqrt{1 + \left(\frac{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d8d0c97-e66d-424a-9c52-d4c7d8464ad8%2F4f909fb1-c6ec-4ab8-8ea2-b6e378bcf41b%2Ffpk73vp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Section 9–24: Curve Length Calculation**
In this section, we'll find the exact length of each given curve. The curves specified are provided along with their respective intervals of \( x \) or \( y \). Each function and its interval are listed below:
1. **Curve 9**:
\[ y = \frac{2}{3}x^{3/2}, \quad 0 \leq x \leq 2 \]
2. **Curve 10**:
\[ y = (x + 4)^{3/2}, \quad 0 \leq x \leq 4 \]
3. **Curve 11**:
\[ y = \frac{2}{3}(1 + x^2)^{3/2}, \quad 0 \leq x \leq 1 \]
4. **Curve 12**:
\[ 36y^2 = (x^2 - 4)^3, \quad 2 \leq x \leq 3, \quad y \geq 0 \]
5. **Curve 13**:
\[ y = \frac{x^3}{3} + \frac{1}{4x}, \quad 1 \leq x \leq 2 \]
6. **Curve 14**:
\[ x = \frac{y^4}{8} + \frac{1}{4y^2}, \quad 1 \leq y \leq 2 \]
7. **Curve 15**:
\[ y = \frac{1}{2} \ln(\sin 2x), \quad \frac{\pi}{8} \leq x \leq \frac{\pi}{6} \]
8. **Curve 16**:
\[ y = \ln(\cos x), \quad 0 \leq x \leq \frac{\pi}{3} \]
### Explanation of Tasks:
1. **Integral Form for Arc Length**:
The exact length \( L \) of a curve given by \( y = f(x) \) over an interval \( [a, b] \) is calculated using the formula:
\[
L = \int_{a}^{b} \sqrt{1 + \left(\frac{
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