Definite integrals & areas
Practice Flashcards
What is ∫ₐᵃ f(x) dx?
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All Flashcards in Topic 5.11
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5.11.19 cards
What is ∫ₐᵃ f(x) dx?
0 — a zero-width interval has zero area.
What happens if you swap the limits of a definite integral?
The sign flips: ∫ₐᵇ f = −∫_b^a f.
State the interval-splitting property.
∫ₐᵇ f + ∫_b^c f = ∫ₐ^c f.
How do you evaluate a definite integral?
Find an antiderivative F, then compute F(b) − F(a).
What mode should the GDC be in for trig integrals?
Radian mode (the limits are in radians).
∫₀^(π/2) cos x dx = ?
[sin x]₀^(π/2) = 1.
What does the integral of a rate of change give?
The total (accumulated) change over the interval.
How do you find the amount at time b from a rate?
amount(b) = amount(a) + ∫ₐᵇ (rate) dt.
∫₀^π sin x dx = ?
[−cos x]₀^π = 2.
5.11.29 cards
What does a negative definite integral tell you about the region?
The region lies below the x-axis; the area is the magnitude of the integral.
Is area ever negative?
No — take the absolute value of a negative integral for the area.
How do you find the area between two curves?
∫ₐᵇ (top − bottom) dx, where 'top' is the upper curve.
How do you decide which curve is the 'top'?
Test an x-value between the limits (or sketch) to see which has greater y.
How do you find the limits for an enclosed region between two curves?
Solve top = bottom to find the intersection x-values.
Area between a curve and the x-axis when it dips below?
Split at the x-intercepts and add the magnitudes (or take |∫|).
Area between y = x and y = x² on [0,1]?
∫₀¹ (x − x²) dx = 1/6.
Why does (top − bottom) work even below the axis?
Subtracting the lower curve measures the vertical gap, which is always positive.
First step for an enclosed area between two curves?
Find the intersection points (solve top = bottom) for the limits.
Topic 5.11 study notes
Full notes & explanations for Definite integrals & areas
Math AA SL exam skills
Paper structures, command terms & tips
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