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.

All Definite Integration notes NDA Maths strategy

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.

Concept 1 of 1

Area between curves

Intuition

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.

Worked example

Find the area enclosed between

y=x^2

and

y=x

Intersections: $x^2=x \Rightarrow x=0,1$ .
On $[0,1]$ , $x \ge x^2$ , so area $=\int_0^1 (x-x^2)\,dx$ .
$=\big[\tfrac{x^2}{2}-\tfrac{x^3}{3}\big]_0^1 = \tfrac12-\tfrac13$ .

Answer:

\frac16

Practice this conceptself-check · 3 quick reps

Try it yourself

Find the area bounded by

y=|x^2-1|

and the x-axis between its roots.

Practice — Level 1 (3 reps)

Quick reps to lock in the method. Try each, then check.

1.
Area between $y=f(x)$ and x-axis on $[a,b]$ ?
2.
How do you find the limits for area between two curves?
3.
Area between $y=4-x^2$ and the x-axis on $[-2,2]$ ?

From the bank · past-year question

Example 1Definite IntegrationMODERATE

Let the function

f(x) = x^2 - 1

What is the area bounded by the function

f(x)

and the

x

-axis?

[Q90 · Apr · 2025]

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.

Watch out for (1)

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

Mastery check — 2 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Definite IntegrationEASY

The slope of the tangent to the curve

y = f(x)

(x, f(x))

is 4 for every real number

x

and the curve passes through the origin.

What is the area bounded by the curve, the

x

-axis and the line

x = 4

[Q78 · Apr · 2025]

Example 2Definite IntegrationMODERATE

Let the curve

f(x) = |x - 3|

What is the area bounded by the curve

f(x)

and

y = 3

[Q86 · Apr · 2025]

Drill every past-year question on this subtopic

3 questions from the bank — paginated, with cart and Word-export support.

Drill the 3 questions