[Diagram: math-integration-area] - Available in full study mode
The definite integral: \intab f(x) dx = [F(x)]ab = F(b) - F(a) where F(x) is any antiderivative of f(x). This equals the signed area between the curve and the x-axis from x = a to x = b. Note: NO +C needed for definite integrals — it cancels out.
Signed vs unsigned area: If the curve goes BELOW the x-axis (f(x) < 0) in the interval, the integral gives a NEGATIVE value. This is called signed area. For geometric area (always positive), you must split the integral at x-intercepts and take absolute value: Area = \intab |f(x)| dx
Area vs integral in IB exams: IB questions are specific: • 'Evaluate the integral' → can be negative, keep the sign • 'Find the area' → always positive, take absolute value if below x-axi • 'Find the area enclosed between the curve and the x-axis' → find roots first, integrate between them, make positive
Area between two curves: To find the area between f(x) (top) and g(x) (bottom) from x = a to x = b: \text(Area) = \intab [f(x) - g(x)] dx The limits a and b are usually the intersection points of the two curve.
Steps for area between curves: 1. Find the intersection points (limits) 2. Determine which curve is on top (test a value between the limits) 3. Integrate [top − bottom] between the limit 4. The answer should be positive — if negative, you have the order wrong
Finding limits from graphs: On Paper 2, intersection points (limits) can be found using the GDC: • Plot both curves and use 'intersection' function • Or solve algebraically For Paper 1, intersection points are always 'nice' numbers — solve algebraically.
Worked example
Apply the key method from Definite Integration and Area Under a Curve in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
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Properties of definite integrals: Key properties for IB: • \intaa f(x) dx = 0 (zero-width interval) • \intab f(x) dx = -\intba f(x) dx (swapping limits negates the integral) • \intab [f(x) + g(x)] dx = \intab f(x) dx + \intab g(x) dx (linearity) • \intac f(x) dx = \intab f(x) dx + \intbc f(x) dx (splitting the interval)
These properties appear as 1-2 mark 'show that' questions: IB uses these properties to create non-computational question: • Given ∫f dx = k, find ∫cf dx or ∫[f + g] dx • Given ∫ over two sub-interval, find ∫ over the full interval Know the splitting property especially well.
Worked example
Apply the key method from Definite Integration and Area Under a Curve in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
Increase in a quantity = definite integral of its rate: If you know the RATE OF CHANGE of a quantity, the increase over an interval [a, b] i: \text(Increase) = \intab (dQ)/(dx) dx = Q(b) - Q(a) This appears in IB a: 'dP/dx is given. Find the increase in profit when x goes from a to b.'
Two ways to find the increase: Method 1 (direct): Integrate the RATE from a to b — ∫[a→b] dP/dx dx Method 2 (after finding P): Calculate P(b) − P(a) Both give the same answer. On Paper 2 you might use GDC for method 1 directly.
'Describe how profit changes' — use f'(x) sign: If asked to 'describe how a quantity changes' over an interval: • Find dQ/dx in that interval • If dQ/dx > 0 → increasing • If dQ/dx < 0 → decreasing • Give a specific reason: 'because dP/dx < 0 for 30 < x ≤ 50'