Integration: all resources · Full notes & worked examples
9709 Pure Mathematics 1 · Maths4U Original
Integration
Learn, then practise
The original worked lessons remain the main explanations. Use these focused reminders to check the conditions before you start.
Power integrals, constants and limits
- Integrate rational powers and linear inner functions with correct constants.
- Evaluate finite integrals and simple improper integrals using valid limits.
- Check singularities and zero-parameter cases before applying a formula.
For \(n\ne-1\), increase the power and divide by the new power. For \((ax+b)^n\), also divide by \(a\), provided \(a\ne0\). A definite integral uses upper value minus lower value. At a singular endpoint or infinity, evaluate a limit; never subtract across an interior singularity.
Read the full notes and 3 worked examples · Next: short practiceRecover a curve from its gradient
- Recover a function from its first or second derivative.
- Determine independent constants using point and slope data.
- Check all conditions and connect the recovered curve to tangents and normals.
Integrate first, then use the given point in the curve equation. Two integrations introduce two independent constants. Slope data belongs in the first derivative. Check the recovered function against every original condition.
Read the full notes and 3 worked examples · Next: short practiceSigned integrals and geometric areas
- Distinguish a signed integral from nonnegative geometric area.
- Find all relevant intersections and split only where the separation changes sign.
- Use vertical or horizontal slices and solve area conditions with parameters.
An integral is signed; geometric area is nonnegative. Find roots or intersections before integrating. Split where the sign or upper curve changes. Use upper minus lower for vertical slices, or right minus left for horizontal slices.
Read the full notes and 3 worked examples · Next: short practiceVolumes of revolution: disks and washers
- Choose the axis, radius, variable and matching limits.
- Use disks or washers as appropriate, without double-counting overlapping rotations.
- Solve inverse volume questions and simple limiting-volume problems.
About the x-axis use \(V=\pi\int y^2\,dx\); about the y-axis use \(V=\pi\int x^2\,dy\). A washer uses \(\pi(R^2-r^2)\), not \(\pi(R-r)^2\). Radii are distances to the axis; check whether the region crosses it.
Read the full notes and 3 worked examples · Next: short practiceAccumulation and mixed calculus review
- Recover amounts from rates and initial values.
- Distinguish net change, final amount and total variation.
- Evaluate variable-limit integrals and check all extrema on a closed interval.
An integral of a rate gives a change: add the initial amount. Find sign changes before calculating total distance or total variation. A variable upper limit can be handled by evaluating the integral explicitly, then differentiating the result.
Read the full notes and 3 worked examples · Next: short practice