Question
Find the area bounded by curves $\left\{(\text{x, y}):\text{y}\geq\text{x}^2 \text{ and y}=|\text{x}|\right\}.$
✓
Answer
The area bounded by the curves, $\left\{(\text{x, y}):\text{y}\geq\text{x}^2 \text{ and y}=|\text{x}|\right\},$ is represented by the shaded region as
It can be observed that the required area is symmetrical about y-axis.
Required area = 2[Area(OCAO) - Area(OCADO)]
$=2\Bigg[\int\limits^1_0\text{x dx}-\int\limits^1_0\text{x}^2\text{dx}\Bigg]$
$=2\bigg[\Big[\frac{\text{x}^2}{2}\Big]^1_0-\Big[\frac{\text{x}^3}{3}\Big]^1_0\bigg]$
$=2\Big[\frac12-\frac13\Big]$
$=2\Big[\frac16\Big]=\frac13\text{units}$
It can be observed that the required area is symmetrical about y-axis.
Required area = 2[Area(OCAO) - Area(OCADO)]
$=2\Bigg[\int\limits^1_0\text{x dx}-\int\limits^1_0\text{x}^2\text{dx}\Bigg]$
$=2\bigg[\Big[\frac{\text{x}^2}{2}\Big]^1_0-\Big[\frac{\text{x}^3}{3}\Big]^1_0\bigg]$
$=2\Big[\frac12-\frac13\Big]$
$=2\Big[\frac16\Big]=\frac13\text{units}$
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