About this topic
Before you begin
Choose substitution, parts or standard integrals, then use exact or numerical methods for bounded areas and volumes of revolution.
Learning goals
- Model and evaluate a nonnegative polynomial curve–line region with vertical strips
- Find complete power antiderivatives on the stated interval
- Evaluate a polynomial definite integral with linked exact working
- Find antiderivatives within the permitted scope
- Evaluate definite integrals
- Relate a definite integral to area under a curve
- Model and evaluate a nonnegative region between polynomial curves with vertical strips
- Find complete logarithmic antiderivatives on the stated interval
- Find complete exponential antiderivatives on the stated interval
- Find complete mixed antiderivatives on the stated interval
- Evaluate definite integrals of powers, reciprocals and exponentials exactly
- 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
- Integrate sin², cos² and tan² of ax + b exactly
- Integrate f′(x)[f(x)]ⁿ, f′(x)/f(x) and f′(x)e^f(x) exactly
- Evaluate definite integrals of trigonometric and inverse-tangent forms exactly
- Select and apply permitted integration techniques
- Model and evaluate a polynomial curve–line region extending below the x-axis
- Model and evaluate a region between polynomial curves extending below the x-axis
- Form polynomial disc and washer models and evaluate their exact volumes
- 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.
Choose a lesson
Work through them in order.
- Not startedAntiderivatives and positive areaNextRecover antiderivatives with their conditions, then select the correct boundaries and vertical separation for positive area.
- Not startedExponential integrals and between-curve areasNextCheck inner factors, exact and numerical integrals, and changing upper/lower boundaries for regions on or above the axis.
- Not startedSubstitution, parts and standard integralsNextChoose and combine integration methods, transform bounds and branches consistently, and verify complete antiderivatives by differentiation.
- Not startedSigned regions and volumes of revolutionNextBuild signed sums, region areas and rotated cross-sections from their boundaries, then verify exact or numerical integrals.