Definite Integrals, Areas and Initial Conditions
Evaluate exact definite integrals, distinguish signed integral from area and reconstruct functions from rates.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
exact limits
Exam cue: Which function is upper over the interval?
Concept 2
fundamental theorem
Exam cue: Does the graph cross the x-axis?
Concept 3
signed area
Exam cue: What initial value determines the integration constant?
Concept 4
area between curves
Concept 5
initial conditions
Risk pitfalls and guardrails
Reversing limits without handling the sign
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Calling a negative definite integral an area
Guardrail: Use correct notation and check coefficients, signs and the constant of integration.
Omitting dx or the constant before using an initial condition
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Upper minus lower
For area between graphs, integrate the top function minus the bottom.
Right value minus left value
Evaluate F(upper limit) − F(lower limit).
Integral is signed
Split at crossings or take appropriate magnitudes for geometric area.
Exact limits stay exact
Retain fractions, surds or radians as required.
Integrate rate to recover amount
A rate function determines a family of original functions.
Initial value chooses the member
Use the condition to find C before the final evaluation.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Evaluate ∫₀² 3x² dx.
Evaluate ∫₁³ 2x dx.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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