Radford Mathematics IB Mathematics resources • AA SL

IB Mathematics: Analysis and Approaches SL — Topic 5 Calculus

Definite Integrals: Evaluation and Properties

Evaluate definite integrals analytically and interpret them as accumulated signed change.

AA SL · SL 5.11

Learning goal

Use the Fundamental Theorem of Calculus form required for definite-integral evaluation and apply basic properties.

Syllabus link

AA SL: SL 5.11 · Topic 5 Calculus.

Big idea

A definite integral is a number. It measures net accumulation and therefore keeps the sign of contributions below the axis.

Key relationship

\(\int_a^b f(x)dx=F(b)-F(a)\)

1

Core idea

A definite integral is a number. It measures net accumulation and therefore keeps the sign of contributions below the axis.

Use the Fundamental Theorem of Calculus form required for definite-integral evaluation and apply basic properties.

2

Key results and method

\[\int_a^b f(x)dx=F(b)-F(a)\]
\[\int_a^a f(x)dx=0\]
\[\int_b^a f(x)dx=-\int_a^b f(x)dx\]
  1. Find an antiderivative \(F\).
  2. Evaluate \(F(b)-F(a)\).
  3. Interpret the sign/context rather than calling every definite integral an area.
3

Worked example

Question

Evaluate \(\int_0^2(x^2+1)dx\).

Solution
  1. An antiderivative is \(F(x)=x^3/3+x\).
  2. \(F(2)-F(0)=8/3+2\).
Answer: \(14/3\).
4

Practice

Try these questions before moving on.

  1. Reverse limits and predict the sign.
  2. Explain why a definite integral can equal zero even when the function is non-zero.
  3. Use additivity to split an integral at \(c\).

Search the worksheet library for more practice →

5

Tutorials and premium resources

▶ Definite integrals as signed area
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