Integration.
Connect antiderivatives, accumulated change, geometric area and motion through a complete integration sequence.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Antiderivatives and constants
Reverse differentiation, keep the arbitrary constant, and check an antiderivative by differentiating it.
- 02
Power/exponential/trigonometric antiderivatives
Integrate powers, natural exponentials and basic trigonometric functions, including linear changes of input.
- 03
Antiderivatives involving reciprocal linear expressions, linked back to logarithmic derivatives
Recognise reciprocal linear expressions and use logarithmic antiderivatives with correct constants and domains.
- 04
Initial-value problems
Use a known value to choose one function from a family of antiderivatives.
- 05
Definite integrals
Evaluate definite integrals and use reversed bounds and linearity correctly.
- 06
Numerical area approximation and the integral idea
Estimate accumulation from rectangles, trapeziums and sampled rates, and distinguish an estimate from an exact integral.
- 07
Fundamental theorem of calculus
Connect an accumulated integral with its rate of change, including a variable upper endpoint.
- 08
Signed area versus geometric area
Calculate geometric area by splitting where a curve crosses the axis, rather than cancelling positive and negative regions.
- 09
Areas between curves
Find intersections, identify the upper curve, and integrate the vertical difference between two graphs.
- 10
Accumulated change
Convert a rate into net change and combine it with a starting amount.
- 11
Displacement and total distance
Separate displacement from total distance by identifying changes of direction from velocity.
- 12
Contextual integral models
Build, evaluate and critique integral models for costs, flows, average values and other contextual quantities.
Topic review
Revisit the key ideas, then choose a practice paper.