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📝 Length Of A Plane Curve

📖 From Calculus • 7. Applications of the Definite Integral In Geometry, Science, and Engineering • 23 questions available

Practice MCQs for Length Of A Plane Curve. Test your knowledge with carefully crafted questions across easy, medium, and hard difficulty levels.

🔄 Last updated: 2026-07-16

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📝 Sample Questions

Q1. What is the general formula for the length of a curve \\(y = f(x)\\) from \\(x = a\\) to \\(x = b\\)?

🔹 A. \\(\\int_{a}^{b} f(x)\\,dx\\)

🔹 B. \\(\\int_{a}^{b} \\sqrt{1+[f'(x)]^{2}}\\,dx\\)

🔹 C. \\(\\int_{a}^{b} |f'(x)|\\,dx\\)

🔹 D. \\(\\int_{a}^{b} (1+f'(x)^{2})\\,dx\\)

💡 Difficulty: easy | ✅ Correct: B

Q2. A roller‑coaster track follows \\(y = 0.1x^{2}\\) for \\(0 \\le x \\le 100\\) meters. Which integral gives its length?

🔹 A. \\(\\int_{0}^{100} \\sqrt{1+0.04x^{2}}\\,dx\\)

🔹 B. \\(\\int_{0}^{100} \\sqrt{1+0.2x}\\,dx\\)

🔹 C. \\(\\int_{0}^{100} \\sqrt{1+0.01x^{2}}\\,dx\\)

🔹 D. \\(\\int_{0}^{100} \\sqrt{1+0.04x}\\,dx\\)

💡 Difficulty: medium | ✅ Correct: A

Q3. A cable follows the catenary \\(y = a\\cosh\\left(\\frac{x}{a}\\right)\\) between \\(x = -L\\) and \\(x = L\\). Which integral gives its total length?

🔹 A. \\(\\int_{-L}^{L} \\sqrt{1+\\sinh^{2}\\left(\\frac{x}{a}\\right)}\\,dx\\)

🔹 B. \\(\\int_{-L}^{L} \\cosh\\left(\\frac{x}{a}\\right)\\,dx\\)

🔹 C. Both A and B are equivalent

🔹 D. \\(\\int_{-L}^{L} \\sqrt{1+\\cosh^{2}\\left(\\frac{x}{a}\\right)}\\,dx\\)

💡 Difficulty: hard | ✅ Correct: C

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🔗 Related Topics

📝 Approximating Roots 📝 Areas and Limits 📝 Continuity in Applications 📝 Continuity of Compositions 📝 Continuity of Inverse Functions

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