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Integration methods, geometric areas and volumes of revolution topic check

Practise integration methods, geometric areas and volumes of revolution in H2 Mathematics, check independently, and return later to review. This check covers selected skills, not the whole course topic.

Learning goals

Produce complete inverse-tangent and inverse-sine standard antiderivatives on stated intervals
Produce complete difference-of-squares logarithmic antiderivatives on stated pole-free intervals
Use a given substitution on stated intervals with linked inverse, differential and complete primitives
Use nontrivial integration by parts with complete linked working and antiderivative families
Find complete power antiderivatives on the stated interval
Find complete logarithmic antiderivatives on the stated interval
Find complete exponential antiderivatives on the stated interval
Find complete mixed antiderivatives on the stated interval
Model and evaluate a nonnegative polynomial curve–line region with vertical strips
Model and evaluate a polynomial curve–line region extending below the x-axis
Model and evaluate a nonnegative region between polynomial curves with vertical strips
Model and evaluate a region between polynomial curves extending below the x-axis
Form polynomial disc and washer models and evaluate their exact volumes
Evaluate a polynomial definite integral with linked exact working

GCE A-Level H2 Mathematics 2027

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Choose a practice setModel and evaluate a nonnegative polynomial curve–line region with vertical strips · 2 questionsModel and evaluate a polynomial curve–line region extending below the x-axis · 2 questionsModel and evaluate a nonnegative region between polynomial curves with vertical strips · 2 questionsModel and evaluate a region between polynomial curves extending below the x-axis · 2 questionsSubstitution and integration by parts · 10 questionsInverse-trigonometric and logarithmic standard integrals · 5 questionsModel and calculate volumes of revolution · 4 questionsAll skills in this check

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Model and evaluate a nonnegative polynomial curve–line region with vertical strips · Check your progress

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Model and evaluate a polynomial curve–line region extending below the x-axis · Check your progress

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Model and evaluate a nonnegative region between polynomial curves with vertical strips · Check your progress

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Model and evaluate a region between polynomial curves extending below the x-axis · Check your progress

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Substitution and integration by parts · Check your progress

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Inverse-trigonometric and logarithmic standard integrals · Check your progress

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Model and calculate volumes of revolution · Check your progress

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Model and evaluate a nonnegative polynomial curve–line region with vertical strips · Check again

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Model and evaluate a polynomial curve–line region extending below the x-axis · Check again

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Model and evaluate a nonnegative region between polynomial curves with vertical strips · Check again

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Model and evaluate a region between polynomial curves extending below the x-axis · Check again

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Substitution and integration by parts · Check again

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Inverse-trigonometric and logarithmic standard integrals · Check again

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Model and calculate volumes of revolution · Check again

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Retention check

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Model and evaluate a nonnegative polynomial curve–line region with vertical strips · Retention check

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Model and evaluate a polynomial curve–line region extending below the x-axis · Retention check

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Model and evaluate a nonnegative region between polynomial curves with vertical strips · Retention check

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Model and evaluate a region between polynomial curves extending below the x-axis · Retention check

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Substitution and integration by parts · Retention check

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Inverse-trigonometric and logarithmic standard integrals · Retention check

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Model and calculate volumes of revolution · Retention check

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Practise: Model and evaluate a nonnegative polynomial curve–line region with vertical strips

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Practise: Model and evaluate a polynomial curve–line region extending below the x-axis

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Practise: Model and evaluate a nonnegative region between polynomial curves with vertical strips

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Practise: Model and evaluate a region between polynomial curves extending below the x-axis

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Practise: Substitution and integration by parts

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Practise: Inverse-trigonometric and logarithmic standard integrals

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Practise: Model and calculate volumes of revolution

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Revision notes

Revise integration

Choose substitution, parts or standard integrals, then use exact or numerical methods for bounded areas and volumes of revolution.

Key ideas

Antiderivatives and positive area
Recover antiderivatives with their conditions, then select the correct boundaries and vertical separation for positive area.
Exponential integrals and between-curve areas
Check inner factors, exact and numerical integrals, and changing upper/lower boundaries for regions on or above the axis.
Substitution, parts and standard integrals
Choose and combine integration methods, transform bounds and branches consistently, and verify complete antiderivatives by differentiation.
Signed regions and volumes of revolution
Build signed sums, region areas and rotated cross-sections from their boundaries, then verify exact or numerical integrals.

What the syllabus asks

From the GCE A-Level H2 Mathematics 2027 syllabus

Include: integration of f′(x)[f(x)]^n including n = −1, f′(x)e^f(x), sin² x, cos² x, tan² x, 1/(a² + x²), 1/√(a² − x²), 1/(a² − x²), 1/(x² − a²); given substitutions; integration by parts. Exclude reduction formulae.
Include: definite integral as limit of sum and area under a curve; evaluating definite integrals; areas bounded by curve and lines parallel to axes, curve and line, or two curves; area below x-axis; volume of revolution about x- or y-axis; approximating integrals with graphing calculator or software. Exclude area and volume of revolution where the curve is defined parametrically.