PorismaSource review

Your source-to-article checklist

Source review · Applications of integration

All source-review topics · Coverage and provenance · Student practice

Every question has a place in the ledger. That does not prove every item is accurate or complete. Use these links to compare the native question, subparts, working, teaching notes, tables and diagrams with your private originals.

“Source comparison recorded” means a visual source-comparison pass is documented. “Mapped” or “generated” describes an ingestion step. Mathematical review, browser interaction tests and settled screenshot review are separate checks.

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15 question records

Optional ticks record your own inspection in this browser only. They do not change the conversion or mathematical-review claims.

08 · Applications of integration

Edition metadata →

08-01 · Rotate the region between a bell curve and a lineQuestion pp.1 / Solution pp.1

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii, iii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

08 App of integration.pdf · p.1
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.1

08-02 · Change the lower boundary at the intersectionQuestion pp.1 / Solution pp.2

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

5 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Do not use one inner-radius formula across the whole interval. The curves exchange roles at x=1.

Essential answer conditions and source qualifications

Inner radius switches from 1/x to1/√(2−x²) at x=1.

Exact private source page references

08 App of integration.pdf · p.1
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.1
08 App of integration sol.pdf · p.2

08-03 · Find the region before selecting radiiQuestion pp.1 / Solution pp.3, 4

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii, iii, iv.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

5 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

08 App of integration.pdf · p.1
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.3
08 App of integration sol.pdf · p.4

08-04 · Subtract the curved core from a cylinderQuestion pp.1 / Solution pp.5

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: main.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Volume

Finite source region 0≤x≤1; the other intersection lies beyond the pole.

Exact private source page references

08 App of integration.pdf · p.1
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.5

08-05 · Use shells or split the horizontal washersQuestion pp.1 / Solution pp.6

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: main.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

08 App of integration.pdf · p.1
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.6

08-06 · Subtract a triangle from an ellipse quadrantQuestion pp.2 / Solution pp.7, 8

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

$a,b>0$ .

Exact private source page references

08 App of integration.pdf · p.2
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.7
08 App of integration sol.pdf · p.8

08-07 · Cancel whole periods before integratingQuestion pp.2 / Solution pp.9

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b, c.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

This asks for signed integration, not total shaded area. Positive and negative lobes cancel.

Essential answer conditions and source qualifications

$a>0$ ; join the given tangent/sine pieces continuously.

Signed integral, not total area.

Exact private source page references

08 App of integration.pdf · p.2
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.9

08-08 · Keep the parameter branch when eliminating tQuestion pp.3 / Solution pp.10, 11

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii, iii, iv.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed with source branch qualification. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Confirmed branch qualification: the printed Cartesian expression is valid only for 0≤t≤π/2. Counterexample t=π gives (x,y)=(π,0), while the printed expression gives x=0. Question PDF p.3 (printed53); solution PDF p.11 explicitly adds the rising-arc restriction.

Essential answer conditions and source qualifications

iii

Valid 0≤t≤π/2 only. On π/2≤t≤π use x=π−arcsin√y−√y.

Rotate the source’s O-to-A region using the rising branch only.

Exact private source page references

08 App of integration.pdf · p.3
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.10
08 App of integration sol.pdf · p.11

08-09 · Use y as the monotone coordinateQuestion pp.3 / Solution pp.11

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: main.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

3 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed with depicted-region qualification. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

The full real-parameter curve has a third intersection at t=7, outside the depicted arc. The article explicitly selects the region shown in the source diagram; the original wording “at two points” is preserved in the PDF.

Essential answer conditions and source qualifications

Area

Source-selected arc −1/2≤t≤1. Full real-parameter curve also meets the line at t=7, giving a different additional region.

Exact private source page references

08 App of integration.pdf · p.3
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.11

08-10 · Carry negative parameter bounds through the substitutionQuestion pp.4 / Solution pp.12, 13

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: limits, area.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

1 page-item record. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

No separate condition recorded. Check the full question, native working and printed assumptions.

Exact private source page references

08 App of integration.pdf · p.4
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.12
08 App of integration sol.pdf · p.13

08-11 · Use unsigned area and subtract the rotating coreQuestion pp.4 / Solution pp.14

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Finite region −π/3≤x≤π/3.

Exact private source page references

08 App of integration.pdf · p.4
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.14

08-12 · Use identities for integration and geometry for the regionsQuestion pp.5 / Solution pp.15, 16

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a, b, c.i, c.ii.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

6 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

On intervals with sin2x≠0.

Exact private source page references

08 App of integration.pdf · p.5
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.15
08 App of integration sol.pdf · p.16

08-13 · Take the infinite limit and use the bowl radiusQuestion pp.6 / Solution pp.18

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: a.i, a.ii, a.iii, b.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed with notation qualification. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Notation qualification in (a)(iii): q is defined in the question as a finite-n integral, while R extends to infinity. The printed “2q” is recorded separately from the total area 2ln2. Question PDF p.6 (printed56), solution p.18.

Essential answer conditions and source qualifications

a(iii)

Printed 2q is the total only if q denotes the limiting area; the question definesq from the finite-n integral.

Exact private source page references

08 App of integration.pdf · p.6
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.17
08 App of integration sol.pdf · p.18

08-14 · Compare a left-endpoint sum with the integralQuestion pp.7 / Solution pp.19

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: sum, inequality.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

1 page-item record.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

2 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

Bound

For positive integer n: rectangles lie strictly below the curve, so Sn<1/ln2.

Exact private source page references

08 App of integration.pdf · p.7
08 App of integration sol.pdf · p.19

08-15 · Sum local area inequalities, then shift the axisQuestion pp.7 / Solution pp.20, 21, 22

Question and explanations → · Complete native source article →

2025 TJC H2 Maths (9758) JC2 MSM · original exercise attribution retained on source pages. Printed/task labels inventoried: i, ii, iii, translation, volume.

Question / subparts

Source comparison recorded

2 page-item records.

Inspect native target → I compared this with the original

Solution steps

Source comparison recorded

4 page-item records.

Inspect native target → I compared this with the original

Teacher notes

Source comparison recorded

3 page-item records.

Inspect native target → I compared this with the original

Tables

No item inventoried — owner check needed

0 page-item records.

Inspect native target → I compared this with the original

Diagrams

Source comparison recorded

4 page-item records. Native SVG generated; rendering is checked separately

Inspect native target → I compared this with the original

Short answer: source-compared
Mathematical correctness: reviewed. Recorded mathematical review is a first pass; it is not independent expert certification.
Final interactions: Question-first question/short-answer pixels reviewed at320px on baseline9f03a4f. This polish pass adds all248 open-guided DOM reflow checks; all31 added given-figure placements inspected at320px; responsive plots and selected updated layouts checked separately. This is not an exhaustive new working/teacher-note pixel audit.
Rendered source content: Settled source-content screenshots reviewed

Essential answer conditions and source qualifications

$n\ge1$ ; unit-width rectangle lies below the decreasing curve.

iii

Positive integer n; right-hand differences telescope.

Rotation volume

Source right-hand region 1≤x≤3; rotate about x=3.

Exact private source page references

08 App of integration.pdf · p.7
08 App of integration.pdf · p.8
08 App of integration sol.pdf · p.20
08 App of integration sol.pdf · p.21
08 App of integration sol.pdf · p.22

An empty omission list is not a completeness certificate. Add any issue you find to the article or page-item record before calling that part finished.