Radford Mathematics IB Mathematics resources • AA SL

IB Mathematics: Analysis and Approaches SL — Topic 5 Calculus

Areas Using Definite Integrals

Turn graph geometry into one or more positive area integrals.

AA SL · SL 5.11

Learning goal

Find areas bounded by curves and the x-axis or between two curves, splitting when sign/order changes.

Syllabus link

AA SL: SL 5.11 · Topic 5 Calculus.

Big idea

Area is always positive. Use absolute value or split the interval wherever the relevant difference changes sign.

Key relationship

\(A=\int_a^b|f(x)|dx\)

1

Core idea

Area is always positive. Use absolute value or split the interval wherever the relevant difference changes sign.

Find areas bounded by curves and the x-axis or between two curves, splitting when sign/order changes.

2

Key results and method

\[A=\int_a^b|f(x)|dx\]
\[A=\int_a^b|f(x)-g(x)|dx\]
\[A=\int_a^b(\text{upper}-\text{lower})dx\quad\text{when order is fixed}\]
  1. Sketch or graph the curves.
  2. Find all intersection/x-intercept limits.
  3. Determine the sign or which curve is upper on each interval.
  4. Integrate positive contributions and add them.
3

Worked example

Question

Find the area enclosed by \(y=4-x^2\) and \(y=x+2\).

Solution
  1. Solve \(4-x^2=x+2\) to get \(x=-2,1\).
  2. The parabola is above the line between the intersections.
  3. Integrate \(\int_{-2}^1[(4-x^2)-(x+2)]dx\).
Answer: \(9/2\) square units.
4

Practice

Try these questions before moving on.

  1. Find the area between \(y=x\) and \(y=x^3\) on \([-1,1]\).
  2. Find total area when a curve crosses the x-axis.
  3. Use GDC absolute-value integration for a non-elementary function.

Search the worksheet library for more practice →

5

Tutorials and premium resources

▶ Area enclosed by a curve and x-axis ▶ Area between two curves — formula ▶ Area between two curves on TI-Nspire
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