Playbook
Applications of Definite Integral
47 q - 1.00/paper - 36% HARD. Effectively one subtopic: Area Bounded by Curves, Axes, and Lines is 43 of its 47 q. A single, well-defined skill, which makes it a cheaper page than its HARD rate suggests.
- Questions in the bank
- 47
- q/paper (2024–25 shifts)
- 1.00
- Tagged HARD
- 36%
- Subtopics
- 2
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
47 q, 1.00/paper, 36% HARD — but effectively a one-skill chapter. Area Bounded by Curves, Axes, and Lines is 43 of the 47 questions, at 33% HARD. The remaining subtopic is 4 questions at 75% HARD and does not justify a place in a study plan.
That concentration makes it a cheaper page than its headline suggests. One skill, learned once, answers 43 of 47 — which is the opposite shape from Limits, where 93 questions are split across two equally hard halves with no cheap entry point.
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 (73 q, 1.85/paper) and Indefinite Integration (159 q, 3.35/paper, 51% 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.
Sketch and intersections
Draw the region, however roughly, and solve the curves simultaneously for the limits. A question is almost never wrong at the integration step and almost always wrong at this one.
Choosing the strip
A vertical strip integrates in x and needs the curves as y in terms of x; a horizontal strip integrates in y. Pick the one that avoids splitting the region.
Top minus bottom
The integrand is upper curve minus lower curve over the interval, or right curve minus left for a horizontal strip. Order matters: reversing it gives the negative of the area.
Splitting at a crossover
Where the curves swap places inside the interval, break the integral at the crossing point and take each piece with its own top curve.
Standard regions and symmetry
Circle, parabola, ellipse and line combinations recur. Exploiting symmetry — computing a quarter or half and multiplying — is usually faster than 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.
Drill every applications of definite integral question
47 questions from the bank, scoped to 2 bundled subtopics.
Drill one subtopic at a time
The 2 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.