Integration Overview

Use this page as the chapter map for the current integration sequence. The aim is for students to move from the meaning of a definite integral to exact area, area between curves, and numerical approximation with the trapezoidal rule.

IB MathematicsCalculusChapter mapOnline textbookRadford Mathematics

On this page

  1. Chapter pathway
  2. Core idea
  3. Current pages
  4. How to study the sequence
  5. Next pages to build

Chapter pathway

The main thread is: interpret the definite integral, use its properties, calculate exact area, and then approximate area numerically with the trapezoidal rule when appropriate.

Chapter mapIntegration overviewThe page that connects the integration lessons. FoundationProperties of definite integralsSigned area, reversing limits, splitting intervals, linearity and symmetry. Application 1Area under a curveUse a definite integral to find area relative to the x-axis. Application 2Area between two curvesUse upper minus lower and split where the upper curve changes.Numerical methodTrapezoidal ruleApproximate area under a curve using equal-width trapezia.

1. The core idea

A definite integral adds infinitely many very thin vertical strips. For area under a curve, the strip height is usually the value of the function. For area between two curves, the strip height is the difference between the upper function and the lower function.

\[\int_a^b f(x)\,dx\]

This expression gives signed area. Regions above the x-axis contribute positively and regions below the x-axis contribute negatively. When a question asks for total area, students often need to split the interval so each physical area is counted as positive.

2. Current integration pages

FoundationProperties of definite integralsStudents learn the rules that justify reversing limits, splitting intervals, adding integrals, taking constant multiples outside, and using symmetry. Area applicationArea under a curveStudents connect definite integrals to vertical strips between a curve and the x-axis. Area applicationArea between two curvesStudents extend the strip idea so the height is upper curve minus lower curve.Numerical methodTrapezoidal ruleStudents approximate area from function values or table data using equal-width trapezia.

3. How to study this sequence

  1. Start with the properties page and make sure the difference between signed area and total area is clear.
  2. Move to area under a curve and practise finding intercepts, choosing limits and interpreting regions above or below the x-axis.
  3. Move to area between two curves and practise finding intersections and deciding which function is upper or lower.
  4. Use the trapezoidal rule when a question asks for an approximation, especially from table data.
  5. When the upper curve or the sign of the function changes, split the interval before integrating.

4. Next page to build

The trapezoidal-rule lesson is now part of this mini-chapter. The next natural page to build is kinematics with integration, connecting velocity-time graphs, displacement and total distance.

← Notes indexProperties of definite integralsTrapezoidal rule →