Application of Integrals — Part 1: Areas under lines and parabolas: visual lesson, CBSE Class 12 Mathematics

Lesson

Application of Integrals, Part 1

A definite integral adds up thin strips: that is exactly an area.

In this lesson you will

Find the area under a curve as a definite integral
Handle curves below or crossing the x-axis
Use horizontal strips for regions beside the y-axis
Find areas bounded by parabolas in standard form and lines parallel to an axis
Use symmetry to shorten the work

The shaded region under $x^2 = 3y$ from $x = 0$ to $x = 3$ has area $3$ sq units.

Concept

What you will be able to do

Write an area as a definite integral
Handle curves below or crossing the $x$ -axis
Slice with horizontal strips when the region sits beside the $y$ -axis
Find areas inside parabolas cut off by a line parallel to an axis
Use symmetry to halve the work

Two ways to slice a region into thin strips.
Strip	Width	Length	Area of the region
vertical	$dx$	$y = f(x)$	$\displaystyle\int_a^b y\,dx$
horizontal	$dy$	$x = g(y)$	$\displaystyle\int_c^d x\,dy$

Concept

Area as a sum of strips

Cut the region under $y = f(x) \ge 0$ into vertical strips of width $dx$ and height $y$ . Each strip has area $y\,dx$ ; adding them all from $x = a$ to $x = b$ gives the area.

A = \int_a^b y\,dx = \int_a^b f(x)\,dx

Two ways to slice a region into thin strips.
Strip	Width	Length	Area of the region
vertical	$dx$	$y = f(x)$	$\displaystyle\int_a^b y\,dx$
horizontal	$dy$	$x = g(y)$	$\displaystyle\int_c^d x\,dy$

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