Application of Integrals — Part 1: Area bounded by a curve: visual lesson, ISC Class 12 Mathematics

Lesson

Application of Integrals, Part 1

A definite integral of a non-negative function measures the area under its graph.

In this lesson you will

Find the area between a curve and the x-axis using a definite integral
Handle curves that lie below or cross the x-axis
Use horizontal strips for curves of the form x = g(y)
Find areas under modulus, exponential and logarithmic curves, and of circles and ellipses

Part 1 at a glance.

Concept

The formulas of Part 1.
Region	Area
Under $y = f(x) \ge 0$ , from $x = a$ to $x = b$	$\displaystyle\int_a^b f(x)\,dx$
Between $y = f(x) \le 0$ and the $x$ -axis	$\left\|\displaystyle\int_a^b f(x)\,dx\right\|$
Between $x = g(y) \ge 0$ and the $y$ -axis, $c \le y \le d$	$\displaystyle\int_c^d g(y)\,dy$
Ellipse with semi-axes $a$ , $b$	$\pi ab$

Diagram

Area from $x = 0$ to $x = 2$: $\int_0^2 \left(\tfrac{x^2}{2} + 1\right)dx = \tfrac{10}{3}$.

If $f(x) \ge 0$ on $[a, b]$ , the area under $y = f(x)$ from $x = a$ to $x = b$ is $\displaystyle\int_a^b f(x)\,dx$ .

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