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ACARA ACMMM131, Fundamental Theorem: evaluating definite integrals

Year 11-12 · Mathematical Methods

Use the rule that the definite integral of f(x) from a to b equals F(b) - F(a), where F is an anti-derivative of f, to evaluate definite integrals.

Below are free printable maths worksheets that support ACMMM131. Print them as PDFs, or generate a fresh one. Every maths answer key is computed in code, so it is never wrong.

Teaching guide

What students learn

Use the rule that the definite integral of f(x) from a to b equals F(b) - F(a), where F is an anti-derivative of f, to evaluate definite integrals.

How to teach it

Use the Fundamental Theorem: find any antiderivative F, then compute F(b) - F(a). Lay the work out as [F(x)] from a to b and subtract. Keep exact fractions rather than rounding.

Worked example
Evaluate the definite integral of (-4x^2 + 5x + 4) dx from x = -2 to x = 1.
Answer: -15/2
Why: Antiderivative F(x) = -(4/3)x^3 + (5/2)x^2 + 4x. Evaluate F(1) - F(-2) = 31/6 - (38/3) = -15/2.

Generated and checked in code, so it is correct.

Watch out for

Swapping the limits (F(a) - F(b)), sign slips when the lower limit is negative, and dropping the brackets so only part of F(a) is subtracted.

Make a worksheet for ACMMM131
More Year 11-12 Mathematical Methods codes: ACMMM114 · ACMMM116 · ACMMM117 · ACMMM121 · ACMMM122 · ACMMM125 · ACMMM132
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Australian Curriculum content descriptions (c) ACARA, used under CC BY 4.0. Descriptions are paraphrased. See australiancurriculum.edu.au.