📝 Integral Curves

📖 From Calculus • 6. Integration • 21 questions available

Practice MCQs for Integral Curves. 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 an integral curve of a first‑order differential equation \(\frac{dy}{dx} = f(x,y)\)?

🔹 A. A: A curve that maximizes the function \(f\)

🔹 B. B: A curve whose tangent at each point satisfies \(\frac{dy}{dx} = f(x,y)\)

🔹 C. C: Any curve passing through the origin

🔹 D. D: A curve that is orthogonal to the direction field

💡 Difficulty: easy | ✅ Correct: B

Q2. A particle moves in the plane with velocity field \(\mathbf{v}(x,y) = (-y, x)\). Which family of curves does its trajectory (integral curve) follow?

🔹 A. A: Straight lines through the origin

🔹 B. B: Circles centered at the origin

🔹 C. C: Parabolas opening upward

🔹 D. D: Ellipses with axes aligned with coordinate axes

💡 Difficulty: medium | ✅ Correct: B

Q3. Consider the nonlinear ODE \(y' = y^2 - x^2\). Which statement correctly describes the qualitative behavior of its integral curves?

🔹 A. A: All curves eventually become unbounded as \(x \to \infty\)

🔹 B. B: Curves intersect the line \(y = x\) only at isolated points

🔹 C. C: Integral curves are symmetric with respect to the line \(y = -x\)

🔹 D. D: There exists a family of curves that remain bounded for all \(x\)

💡 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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