Intro
Integration solves geometry problems by summing infinitesimal strips — areas between curves, volumes of revolution, and arc-length style set-ups in Extension 2. Sketch the region first and label intersection points before setting bounds. Keywords: integration applications, HSC calculus, geometry and integration. NSW HSC Mathematics.
Summary
For areas: find intersections, identify upper curve, integrate (top − bottom). For volumes of revolution: state axis clearly and use π∫[f(x)]² dx. Substitution and by-parts appear on composite boundaries.
Integration geometry questions frequently combine multiple techniques — substitution on one interval, by parts on another. Label each integral separately before combining results for the final answer.
Key Points
- Area between curves: intersection points define limits; subtract lower from upper.
- Horizontal strips sometimes simplify when functions are inverse pairs.
- Volume of revolution: disk/washer about x- or y-axis — label axis explicitly.
- Sketch region before integrating — prevents reversed bounds.
- Check answers with symmetry or rough estimate.
- Techniques in HSC Integrals booklet.
Worked example
Question. Find the area bounded by y = x and y = x² between x = 0 and x = 1.
Solution.
- On [0, 1], x ≥ x² (line above parabola near origin).
- Area = ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.
Answer. Area = 1/6 square units.
Takeaway. Confirm which curve is upper on the interval — swap if integral would be negative.
Exam Preparation
Integration geometry is mark-heavy in Extension 1 and 2. Practise sketching, finding intersections, and setting bounds without integrating first. Extension 2 students should add one volume-of-revolution question weekly.
Volume questions sometimes specify rotation about a line not on the coordinate axis — translate coordinates or use the formula your syllabus provides. Sketch the solid in 3D even roughly — it confirms whether washers or disks apply.
- Intersection drills. Find points for new curve pairs without integrating.
- Area vs volume. Alternate question types to avoid formula mix-ups.
- Bound checks. Verify upper minus lower is positive on each interval.
Area questions with curves defined piecewise require splitting integrals at boundary points. Volumes about the y-axis may use shells or washers depending on syllabus emphasis — follow your teacher's preferred method consistent with past papers. Extension 2 may include solids with known cross-sections; integrate the area function A(x) along the axis. Always draw and shade the region — it catches swapped bounds before you integrate. Show every integration step for method marks even when the antiderivative is routine.
Mini-FAQ
What if curves cross mid-interval?
Split the integral at intersection points where upper/lower swap.
Shell method in HSC?
Disk/washer about axes is standard — follow your syllabus and past papers for expected methods.
Do I need exact answers?
Give exact unless rounding is specified — fractions and π are preferred over decimals.
Common mistakes to avoid
- Integrating bottom minus top (negative area).
- Wrong limits from algebra errors finding intersections.
- Volume formula without stating axis of rotation.
- Forgetting π in volume of revolution.
Integration geometry checklist: (1) sketch region, (2) find intersections, (3) identify upper minus lower, (4) set bounds, (5) integrate, (6) check sign. Apply the checklist to every area question until automatic — then add volume of revolution with axis stated explicitly. Split integrals at any point where upper and lower curves swap.
Areas and volumes often appear in the same Extension paper — do not confuse disk volume formulas with area integrals. Extension 2 may ask for regions bounded by curves and lines — find all intersection points before integrating.
Estimate area with a rough sketch before integrating — catches reversed bounds immediately. Volume of revolution questions punish ambiguous axes of rotation; write 'rotating about the x-axis' before setting up π∫[f(x)]² dx. Split integrals wherever upper and lower curves swap; piecewise boundaries are common in Extension 2 geometry questions. Check answers with symmetry when regions are symmetric about an axis. The Integrals booklet on vumaths.com groups area and volume examples by technique. Always state whether disk or washer method applies when rotating about an axis. Substitution often simplifies integrands with composite boundaries — identify the inner function first.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Integrals booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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