Question
Find the area between the curves $y = x$ and $y = x^2.$
✓
Answer
The required area is represented by the shaded area $OBAO$ as
The points of intersection of the curves, $y = x$ and $y = x^2,$ is $A(1, 1).$
We draw $AC$ perpendicular to $x-$axis.
$\therefore$ Area (OBAO) = Area $(\Delta\text{ OCA}) - $ Area $(OCABO) ...(i)$
$=\int\limits^1_0\text{x dx}-\int\limits^1_0\text{x}^2\text{dx}$
$=\Big[\frac{\text{x}^2}{2}\Big]^1_0-\Big[\frac{\text{x}^3}{3}\Big]^1_0$
$=\frac12-\frac13$
$=\frac16\text{ units}$
The points of intersection of the curves, $y = x$ and $y = x^2,$ is $A(1, 1).$
We draw $AC$ perpendicular to $x-$axis.
$\therefore$ Area (OBAO) = Area $(\Delta\text{ OCA}) - $ Area $(OCABO) ...(i)$
$=\int\limits^1_0\text{x dx}-\int\limits^1_0\text{x}^2\text{dx}$
$=\Big[\frac{\text{x}^2}{2}\Big]^1_0-\Big[\frac{\text{x}^3}{3}\Big]^1_0$
$=\frac12-\frac13$
$=\frac16\text{ units}$
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