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📝 Area Of A Surface Of Revolution

📖 From Calculus • 7. Applications of the Definite Integral In Geometry, Science, and Engineering • 21 questions available

Practice MCQs for Area Of A Surface Of Revolution. Test your knowledge with carefully crafted questions across easy, medium, and hard difficulty levels.

🔄 Last updated: 2026-07-16

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Q1. What is the general formula for the surface area generated by rotating \(y=f(x)\) about the \(x\)-axis on the interval \([a,b]\)?

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

🔹 B. \(2\pi\int_{a}^{b}f(x)dx\)

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

🔹 D. \( \pi\int_{a}^{b}[f(x)]^{2}dx\)

💡 Difficulty: easy | ✅ Correct: A

Q2. When rotating the curve \(y=\sin x\) about the \(x\)-axis from \(0\) to \(\pi\), which integral correctly represents the surface area?

🔹 A. \(2\pi\int_{0}^{\pi}\sin x\sqrt{1+\cos^{2}x}\,dx\)

🔹 B. \(2\pi\int_{0}^{\pi}\sin x\sqrt{1+\sin^{2}x}\,dx\)

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

🔹 D. \( \pi\int_{0}^{\pi}\sin^{2}x\,dx\)

💡 Difficulty: medium | ✅ Correct: A

Q3. A spacecraft component is created by rotating \(y=\sqrt{1+x^{3}}\) about the \(x\)-axis from \(x=0\) to \(x=2\). Which integral correctly represents its surface area?

🔹 A. \(2\pi\int_{0}^{2}\sqrt{1+x^{3}}\sqrt{1+\frac{9x^{4}}{4(1+x^{3})}}dx\)

🔹 B. \(2\pi\int_{0}^{2}\sqrt{1+x^{3}}\sqrt{1+\frac{9x^{4}}{4}}dx\)

🔹 C. \(2\pi\int_{0}^{2}(1+x^{3})\sqrt{1+\frac{9x^{4}}{4(1+x^{3})}}dx\)

🔹 D. \( \pi\int_{0}^{2}(1+x^{3})dx\)

💡 Difficulty: hard | ✅ Correct: A

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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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