NDA Maths · Definite Integration

Area Under and Between Curves

Area is the integral of the gap between curves; use the absolute value (or split at intersections) so every piece counts positively, and exploit symmetry to halve the work.

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Why this matters

A small but recurring subtopic (3 PYQs). The recipe is fixed: find the intersection points (the limits), integrate top-minus-bottom, and take the magnitude so area never comes out negative.

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Concept 1 of 1: Area between curves

Area is always positive, so set up the integral of the upper curve minus the lower one between their intersection points. When a curve dips below the axis, the |·| (or a split) keeps the area positive; symmetry about a line lets you integrate half and double.

Definition

The area recipe:

Area between $y=f(x)$ and the x-axis on $[a,b]$ is $\displaystyle\int_a^b |f(x)|\,dx$ .
Area between two curves is $\displaystyle\int_a^b |f(x)-g(x)|\,dx$ , with $a,b$ the intersection points (solve $f=g$ ).
Use symmetry: if the region is symmetric about a vertical line, integrate one half and double.

Area as a definite integral

A = \int_a^b |f(x)|\,dx \qquad A = \int_a^b |f(x)-g(x)|\,dx

Worked example

Find the area enclosed between

y=x^2

and

y=x

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

NDA · Apr 2025 · Q90Moderate

Example 1 · Definite Integration · Area Under Curves

Let the function

f(x) = x^2 - 1

What is the area bounded by the function

f(x)

and the

x

-axis?

Area is unsigned — use the absolute value

Plain

\int_{-1}^{1}(x^2-1)\,dx

gives a NEGATIVE number, which cannot be an area. Take

|f|

(or split at the roots and add magnitudes) so the region below the axis still adds positively.

Drill 2 more on area between curves

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Area between curves
Area as a definite integral
$A = \int_a^b |f(x)|\,dx \qquad A = \int_a^b |f(x)-g(x)|\,dx$

Watch out for (1)

Area is unsigned — use the absolute value
→ Area between curves

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