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Application of Integrals

Area between curves, every time: sketch first, then integrate top minus bottom. It has grown since the early papers.

Questions in the bank
117
q/paper in 2025–26
1.03
Numeric answer
35%
Notes pages
7

Tier: Core

When you’ll see it

The area of a region bounded by curves and lines, usually one that must be sketched before any integral is written.

How this chapter is tested

Every question is the same two steps: sketch the region, then integrate top minus bottom in x or right minus left in y. The integral is rarely hard. The time goes on the sketch — where the curves meet, which one is on top, and where the top changes.

Parabolas against lines are the core, with vertical strips for curves like y = x² and horizontal strips for sideways parabolas like y² = 4ax. Then come circles and ellipses cut by a parabola or a line, modulus curves that fold into V shapes, and boundaries set by the min or max of two curves. Trigonometric, exponential and reciprocal curves put a trig value or a log into the answer.

The last page turns the question round: the area is given and a constant in a curve is asked. It is the same integral followed by an equation. The chapter leans on Definite Integration for the evaluation and on Conic Sections for the curves; πr², πab and the area of a circular segment save an integral when they fit.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Vertical strips

    Area = ∫ (top − bottom) dx between the meeting points; split the interval where the top or bottom changes.

  • Horizontal strips

    For a sideways parabola, integrate right minus left in y, with limits taken from the y-values of the meeting points.

  • Circles, ellipses and other conics

    Use πr², πab or a segment area for the standard pieces; integrate only what no formula covers.

  • Modulus curves

    Solve each arm of a V separately; |f(x)| reflects the part of f below the x-axis upward.

  • Max, min and piecewise boundaries

    The boundary follows the lower or higher of two curves and switches where they cross; [x²] jumps at x = 1, √2, √3.

  • Trigonometric, exponential and reciprocal curves

    sin x and cos x cross where tan x = 1; ∫ aˣ dx = aˣ/ln a; y = k/x gives a log.

  • Unknown parameters

    Write the area in terms of the constant, set it equal to the value given, and keep the root the conditions allow.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Signed integral for area

    ∫ sin x dx from 0 to 2π is 0, but the area is 4. Split wherever the curve crosses the axis and add the pieces as positive numbers.

  • Limits in the wrong variable

    For horizontal strips the limits are y-values. Using the x-values of the meeting points gives a different region.

  • The lower branch

    y² = kx also has y = −√(kx). If the region is not limited to y ≥ 0, include the part below the axis or double by symmetry.

  • Minor segment for major

    The segment formula gives the smaller piece. When the larger portion is asked, subtract from πr².

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Application of Integrals notes

Drill every Application of Integrals question

117 questions from the bank, across 7 subtopics.

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The 7 subtopics, in teaching order.

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