IB Mathematics AA HL Flashcards SL 5.11 Definite integrals

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[h]SL 5.11 Definite integrals

[q] Definite Integrals/Area

[a]

Manual Process
We saw in 5.5 what a definite integral is, and how to find it, and the area under the curve, by GDC. In this section, we will see how to do it by hand (for paper 1).

If \( F'(x) = f(x) + c \), then \(\int_a^b f(x) dx = F(b) – F(a)\) or simply:

\[
\int_a^b f'(x) dx = g(b) – g(a)
\]

[q]
Meaning: Integrate, plug in \(b\), plug in \(a\), find the difference.

Example 1: \(\int_0^\pi \sin x dx = [-\cos x]_0^\pi = (-\cos \pi) – (-\cos 0) = -(-1) + 1 = 1 + 1 = 2\).

[a]
Area Between Curves
We can find this by using the area under \(f(x)\) minus the area under \(g(x)\), leaves us with the area between the curves. We can either do 2 separate integrals or subtract the functions, simplify, then integrate.

– Example 2: Find the area enclosed by \( f(x) = x^3 – 4x^2 + x + 6 \) and the x-axis:
– Roots at \( x = -1, 2, 3 \). Split the integral and use absolute values:
\[
\text{Area} = \left| \int_{-1}^2 (x^3 – 4x^2 + x + 6) dx \right| + \left| \int_2^3 (x^3 – 4x^2 + x + 6) dx \right|
\]
– After calculating:
\[
\text{Area} \approx 11.8
\]

[q]

Crossing the X-axis
When doing area under curves, we need to be careful of regions below the x-axis, as the integral will naturally give you negative values for the area, which is clearly wrong. So if we take absolute values of each region between roots (or \(\int_a^b |f(x)| dx\) on GDC), we can ensure a correct area.

 

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IB Mathematics AA HL Flashcards SL 5.11 Definite integrals

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