Integration with Initial Conditions
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Flip to reveal answersWhat is an initial condition in integration?
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All 8 Flashcards — Integration with Initial Conditions
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Question
What is an initial condition in integration?
Answer
A specific point (x₀, y₀) that the function passes through. Used to find the exact value of the constant C.
💡 Hint
Initial condition removes the ambiguity of +C.
Question
f'(x) = 4x − 1, f(2) = 5. Find f(x).
Answer
Step 1: Integrate → f(x) = 2x² − x + C. Step 2: f(2) = 8 − 2 + C = 5 → C = −1. Answer: f(x) = 2x² − x − 1.
💡 Hint
Substitute the point AFTER integrating.
Question
If f'(x) = 6x and the curve passes through (0, 4), what is C?
Answer
Integrate: f(x) = 3x² + C. Substitute (0, 4): 3(0) + C = 4 → C = 4. So f(x) = 3x² + 4.
💡 Hint
Easiest initial condition: use x = 0.
Question
In kinematics, if v(t) = 3t², s(0) = 5, what is s(t)?
Answer
Integrate: s(t) = t³ + C. Use s(0) = 5: C = 5. So s(t) = t³ + 5.
💡 Hint
v = ds/dt so s = ∫v dt + C.
Question
How many initial conditions do you need to fully determine a function after integrating twice?
Answer
Two initial conditions — one for each integration, since each introduces a new constant (C₁ and C₂).
💡 Hint
Each ∫ adds one unknown constant.
Question
a(t) = 10, v(0) = 3, s(0) = 1. Find s(t).
Answer
v(t) = 10t + 3 (use v(0)=3 → C₁=3). s(t) = 5t² + 3t + C₂. Use s(0)=1 → C₂=1. s(t) = 5t² + 3t + 1.
💡 Hint
Integrate twice with separate constants.
Question
What is the "particular solution" vs "general solution" of an integral?
Answer
General solution: f(x) + C (all possible solutions). Particular solution: the specific function once C is found using an initial condition.
💡 Hint
Initial condition converts general → particular.
Question
dy/dx = 3x² + 2x, and y = 10 when x = 1. Find y.
Answer
Integrate: y = x³ + x² + C. Use (1, 10): 1 + 1 + C = 10 → C = 8. So y = x³ + x² + 8.
💡 Hint
Substitute after integrating, not before.
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