NDA Maths · Teaching notes

Applications of Integration — NDA Maths

Applications of Integration is a compact, visual chapter: 25 PYQs span 2017-2026, and almost all of them ask one thing — the AREA of a region in the plane. The integration itself is rarely hard; the marks live in the SETUP. You win them by sketching the region, reading the boundary curve and the limit lines off the question, and choosing the right model: area under one curve, area between two curves, or a known shape you never integrate at all. The notes teach in two movements, foundations first: (1) Area Bounded by a Curve, Lines & Axes — the definite integral as signed area, then area under a curve, the below-axis and factor-of-2 traps, polygonal regions from modulus boundaries, the parabola-latus-rectum area, step functions, and circular segments; (2) Area Between Two Curves & Intersection Points — finding where curves meet, the top-minus-bottom integral, curve-versus-line regions, and composite regions built by subtracting known areas. The recurring lesson across both: get the picture right, count the sign, and don't integrate what you can recognise. Every PYQ is tagged.

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Subtopic notes

PYQ weightage by concept

11 concepts · 25 PYQs — where the marks actually sit, so you know what to drill first

Area Bounded by a Curve, Lines & Axes16 PYQs · 64%

Concept	PYQs	Share
Regions Bounded by Lines & Modulus	4	16%
Area Under a Curve Between Two Lines	3	12%
Below the Axis, Loops & the Factor of 2	3	12%
Area of a Parabola Cut by Its Latus Rectum	3	12%
Area of a Circular Segment by a Chord	2	8%
Area Under a Step (Greatest-Integer) Curve	1	4%
The Definite Integral as Signed Areafoundation	—	—

Area Between Two Curves & Intersection Points11 PYQs · 44%

Concept	PYQs	Share
Area Between a Curve and a Line	4	16%
Area Between Curves: Top Minus Bottom	3	12%
Finding Where Two Curves Meet	2	8%
Composite Regions: Subtract Areas	2	8%

Formula & revision sheet

11 formulas · 10 gotchas across all subtopics — the exam-eve cheat-sheet

Area Bounded by a Curve, Lines & Axes

Formulas (7)

The Definite Integral as Signed Area · Area under a curve above the axis $A = \int_a^b f(x)\,dx \quad (f \ge 0)$
Area Under a Curve Between Two Lines · Semicircle area shortcut $y = \sqrt{r^2 - x^2}\ \Rightarrow\ A = \tfrac{1}{2}\pi r^2$
Below the Axis, Loops & the Factor of 2 · Area with a sign change at c $A = \left|\int_a^c f\,dx\right| + \left|\int_c^b f\,dx\right|$
Regions Bounded by Lines & Modulus · Polygon area, not an integral $\text{rectangle } = \text{w}\times\text{h} \qquad \text{triangle } = \tfrac{1}{2}\,b\,h$
Area of a Parabola Cut by Its Latus Rectum · Parabola–latus rectum area $y^2 = 4ax:\quad A = 2\int_0^{a}\sqrt{4ax}\,dx = \tfrac{8}{3}a^2$
Area Under a Step (Greatest-Integer) Curve · One step = one rectangle $A = |n| \times (\text{interval width}), \quad [x] = n$
Area of a Circular Segment by a Chord · Segments of a circle $A_2 = \text{sector} - \text{triangle}, \qquad A_1 = \pi r^2 - A_2$

Watch out for (6)

Integrate only where the curve stays above the axis
→ Area Under a Curve Between Two Lines
The raw integral can be zero while the area is not
→ Below the Axis, Loops & the Factor of 2
|x| ≤ p gives a side of length 2p, not p
→ Regions Bounded by Lines & Modulus
Double the half-region, and use the right limit
→ Area of a Parabola Cut by Its Latus Rectum
Negative step values still give positive area
→ Area Under a Step (Greatest-Integer) Curve
Subtract the triangle from the sector
→ Area of a Circular Segment by a Chord

Area Between Two Curves & Intersection Points

Formulas (4)

Finding Where Two Curves Meet · Intersection condition $f(x) = g(x) \ \Rightarrow\ \text{the } x\text{-values where the curves cross}$
Area Between Curves: Top Minus Bottom · Area between curves $A = \int_a^b \bigl(\text{top} - \text{bottom}\bigr)\,dx$
Area Between a Curve and a Line · Curve over a line $A = \int_a^b \bigl(y_{\text{curve}} - y_{\text{line}}\bigr)\,dx$
Composite Regions: Subtract Areas · Quarter-circle minus a curve $A = \tfrac{1}{4}\pi r^2 - \int_a^b f(x)\,dx$

Watch out for (4)

A modulus can create extra intersections
→ Finding Where Two Curves Meet
Subtract top minus bottom, not in equation order
→ Area Between Curves: Top Minus Bottom
Pick the correct branch of a sideways parabola
→ Area Between a Curve and a Line
Subtract the area under the curve, not the curve's value
→ Composite Regions: Subtract Areas