Definite integration, areas and the trapezium rule flashcards

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Definite integral
An integral with limits: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′(x) = f(x). Its value is a number.
Lower and upper limits
In ∫ₐᵇ f(x) dx, a is the lower limit and b the upper limit; substitute both and subtract, upper minus lower.
Ordinates (trapezium rule)
The y-values y₀, y₁, …, yₙ at the equally spaced x-values a, a + h, …, b.
Strip width h
h = (b − a)/n, where n is the number of strips. n strips use n + 1 ordinates.
Percentage error
|estimate − exact value| ÷ exact value × 100%.
Convex / concave (bending up / down)
Convex: the curve bends upwards and chords lie above it. Concave: it bends downwards and chords lie below it.
∫ₐᵇ f(x) dx
[F(x)]ₐᵇ = F(b) − F(a), where F is any integral of f. No + c. (No — learn it)
Area between a curve and the x-axis (curve above the axis)
Area = ∫ₐᵇ y dx. (No — learn it)
Area between two graphs meeting at x = a and x = b
∫ₐᵇ (y_top − y_bottom) dx. (No — derive it)
∫xⁿ dx (power rule, n ≠ −1)
xⁿ⁺¹/(n + 1) + c: raise the power by one and divide by the new power. In a definite integral the + c cancels. (No — learn it)
Trapezium rule
∫ₐᵇ y dx ≈ ½h{y₀ + yₙ + 2(y₁ + y₂ + … + yₙ₋₁)}, h = (b − a)/n. (Yes)
∫ₐᵇ f(x) dx with the limits swapped
Swapping the limits changes the sign: the integral from b to a equals −∫ₐᵇ f(x) dx. (No — derive it)
Sign of ∫ₐᵇ y dx for a region below the x-axis
Negative. The area is the size of the integral.
A curve crosses the x-axis between the limits
Split the integral at the crossing point; the parts have opposite signs and would partly cancel.
Region bounded by a curve, a straight line and the x-axis
Usually split it: area under the curve (integrate) plus a triangle or trapezium under the line (geometry).
Trapezium rule on a curve that bends upwards
The chords lie above the curve: the rule overestimates.
Trapezium rule on a curve that bends downwards
The chords lie below the curve: the rule underestimates.
Effect of doubling the number of strips
The chords hug the curve more closely; the error becomes smaller (roughly a quarter for smooth curves).
Why is there no + c in a definite integral?
It appears in both F(b) + c and F(a) + c, so it cancels on subtraction.
Can a definite integral be zero when there is a region?
Yes — equal areas above and below the x-axis cancel. Check where the curve crosses the axis.
Why does 'top − bottom' work even below the x-axis?
The vertical gap y_top − y_bottom is positive wherever the top graph is above, whatever the sign of y.
Where do the limits for an area come from?
From the region: where the curve meets the axis, where two graphs meet, or given lines x = a and x = b.
Why does the trapezium rule count inner ordinates twice?
Each inner ordinate is a side of two neighbouring trapezia.
When can you find a percentage error for a trapezium estimate?
Only when the exact value is known, e.g. when the integral can also be found algebraically.
∫₁² (4x³ − 2x) dx
[x⁴ − x²]₁² = (16 − 4) − (1 − 1) = 12.
∫₄⁹ 1/√x dx
[2√x]₄⁹ = 6 − 4 = 2.
Area enclosed by y = 4x − x² and the x-axis
Limits 0 and 4: [2x² − x³/3]₀⁴ = 32 − 64/3 = 32/3.
∫₀¹ (x² − 1) dx, and the area it represents
[x³/3 − x]₀¹ = −2/3; the region is below the axis, so its area is 2/3.
Area between y = x² and y = 2x
Meet at x = 0, 2; line on top: ∫₀² (2x − x²) dx = 4 − 8/3 = 4/3.
Trapezium rule, 2 strips: ∫₀¹ 1/(1 + x) dx
h = 0.5; y = 1, 2/3, 1/2: 0.25{1 + 0.5 + 2(2/3)} = 17/24 ≈ 0.708.
Estimate 4.2, exact value 4: percentage error?
(4.2 − 4)/4 × 100 = 5%.
Given ∫₀² f(x) dx ≈ 3.5, estimate ∫₀² (f(x) + 2) dx
3.5 + ∫₀² 2 dx = 3.5 + 4 = 7.5.
Subtracting F(a) without brackets
Write (F(b)) − (F(a)); every term of F(a) changes sign.
Giving a negative area
Areas are positive. A negative integral means the region is below the axis — state its size.
Integrating straight across a root for an area
Gives the difference of the areas, not their sum. Split at the root.
Using the number of ordinates in h = (b − a)/n
n is the number of strips; there is one more ordinate than strips.
'Over- or underestimate?' with no reason
Give the shape (bends up/down, chords above/below the curve) and then the conclusion.
Rounding ordinates too early
Keep at least 4 decimal places in the table; round only the final estimate.

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