Playbook
Applications of Definite Integral
44 q - 1.04/paper - 32% HARD. Three pages: area under one curve (14 q, 14% HARD - the cheapest page in the strand), area between two curves once the intersections are found (21 q, 38%), and the circle, ellipse and hyperbola regions that need the standard root integrals (9 q, 44%). One well-defined skill, which makes it a cheaper page than its HARD rate suggests.
- Questions in the bank
- 44
- q/paper (2024–25 shifts)
- 1.04
- Tagged HARD
- 32%
- Subtopics
- 3
Strand: Long Tail
When you’ll see it
The word area, together with two curves — or a curve, an axis and a pair of bounding lines.
How this chapter is tested
44 q, 1.04/paper, 32% HARD — but effectively a one-skill chapter. The two area pages — under one curve and between two curves — are 35 of the 44 questions, at 29% HARD. The circle, ellipse and hyperbola page is 9 questions at 44% HARD and needs exactly one standard result learnt cold.
That concentration makes it a cheaper page than its headline suggests. One skill, learned once, answers 35 of 44 — which is the opposite shape from Limits, where 90 questions are spread across seven pages with no cheap one among them.
The skill is not the integration; it is the setup. Find where the curves meet, decide which one is on top over each stretch, decide whether the region is simpler in x or in y, and split the interval wherever the top curve changes. Get that right and what remains is an integral you already know how to do.
It sits directly downstream of Definite Integration (67 q, 1.75/paper) and Indefinite Integration (151 q, 3.42/paper, 52% HARD). If antiderivatives are not fluent this chapter is unreachable; if they are, it is close to free — which is why it belongs late in a plan rather than early.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Area Under a Curve — Between a Curve and an Axis
Sketch, find where the curve meets the axis, and integrate y dx (or x dy for a horizontal strip); where the curve crosses the axis, integrate the modulus piece by piece. Includes the curve-with-unknown-coefficients and the divide-the-area-in-half stems.
Area Between Two Curves — Intersections First
Solve the curves simultaneously for the limits, then integrate upper minus lower (or right minus left for a horizontal strip), splitting wherever the curves swap places. A question is almost never wrong at the integration step and almost always wrong at the intersections.
Areas of Circles, Ellipses and Hyperbolas — Sectors, Segments and Standard Integrals
The integral of sqrt(a^2 - x^2) and sqrt(x^2 - a^2) learnt cold, the sector formula (1/2) r^2 theta, the quarter-ellipse pi ab/4, and symmetry — computing a quarter or half and multiplying — instead of integrating the whole region.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Signed integral offered as area
A region below the x-axis contributes a negative integral. Area needs the magnitude, or a split at the axis crossing; the signed value is on the option list.
A missed intersection
Two curves may meet at more points than the obvious one. A limit taken from the wrong root produces a clean-looking but wrong number.
Curves the wrong way round
Integrating lower minus upper gives the correct magnitude with a minus sign in front, and that negative is a supplied option.
Forcing the wrong variable
A region bounded on the left and right by curves is one integral in y and two or three in x. Choosing x out of habit turns a one-step question into a three-step one, which at 1.8 minutes is the real cost.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Applications of Definite Integral notesDrill every applications of definite integral question
44 questions from the bank, scoped to 3 bundled subtopics.
Drill one subtopic at a time
The 3 subtopics this playbook covers, in catalog order.
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