Find the Area of the region (in square unit) bounded by the curve y = x – 2 and x = 0 to x= 4.

Concept used:

The area between the curves y1 = f(x) and y2 = g(x) is given by:

Area enclosed = \(\rm \int_{x_1}^{x_2} |y_2 - y_1|dx\)

Where x1 and x2 are the intersections of curves y1 and y2

Calculation:

In figure ΔABC and ΔAOD is similar

So, Area of the region = 2 × (Area of ΔABC)

For the area of ΔABC

= \(\rm \mathop \smallint \limits_2^4 \left( {x - 2} \right)\;dx\)

= \(\rm[\frac{x^2} 2 - 2x ]^{4}_2\)     

= \( \frac {4^2}2 - 2(4) - (\frac {2^2}2 - 2\times2)\)

= 8 - 8 - 2 + 4

= 2 sq. unit

So, Area of the region = 2 × (Area of ΔABC) = 2 × 2 = 4 sq. unit

Alternate Method

Using the triange formula = 1/2 × base × height

in the figure = 1/2 × (2 × 2)

Area of ΔABC = 2

Total bounded area = 2 × 2 = 4 sq. unit