Integration
Build and connect the representations and methods in integration.
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Before you begin
Build and connect the representations and methods in integration.
Learning goals
- Find antiderivatives within the permitted scope
- Evaluate definite integrals
- Relate a definite integral to area under a curve
- Select and apply permitted integration techniques
- Relate definite integrals to signed area and geometric area
- Use integrals to calculate volumes of revolution
Syllabus statements covered
- 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.
- 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.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Antiderivatives and positive areaRecover antiderivatives with their conditions, then select the correct boundaries and vertical separation for positive area.
- Exponential integrals and between-curve areasCheck inner factors, exact and numerical integrals, and changing upper/lower boundaries for regions on or above the axis.
- Substitution, parts and standard integralsChoose and combine integration methods, transform bounds and branches consistently, and verify complete antiderivatives by differentiation.
- Signed regions and volumes of revolutionBuild signed sums, region areas and rotated cross-sections from their boundaries, then verify exact or numerical integrals.