Application of Integrals — Part 2: Area between two curves: visual lesson, ISC Class 12 Mathematics

Lesson

Application of Integrals, Part 2

The area between two curves is the integral of the upper curve minus the lower curve.

In this lesson you will

Part 2 at a glance.

Concept

Area between curves = integral of (upper − lower).
Step	What to do
Meeting points	Solve $f(x) = g(x)$ for the limits
Upper curve	Test one point between the limits
Integral	$\displaystyle\int_a^b \big(f(x) - g(x)\big)\,dx$ with $f$ upper
Crossings	If they cross inside $[a, b]$ , split there

Diagram

Area of the lens: $\int_0^2 (2x - x^2)\,dx = \tfrac43$.

Area $= \displaystyle\int_a^b \big(f(x) - g(x)\big)\,dx$ when $f \ge g$ on $[a, b]$ .

\text{Area} = \int_a^b \big(f(x) - g(x)\big)\,dx