Find tha area bounded by the curves x = y^2 and x = 3 - 2y^2. - Vidyadip

Question

Find tha area bounded by the curves $x = y^2$ and $x = 3 - 2y^2.$

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Answer

To find area bounded by
$x = y^2 ...(i)$
and $x = 3 - 2y^2$
$2y^2= -(x - 3) ...(ii)$
Equation $(i)$ represents a parabola with vertex $(0, 0)$ at origin and axis as $x-$axis and equation $(ii)$ represents a line vertex $(3, 0)$ parallel to $y-$axis.
A rough sketch of the equations is as below:
Requried area $=$ Region $\text{OABCO}$
$=2\Big[\int\limits_{0}^{1}\text{y}_{1}\text{ dx}+\int\limits_{1}^{3}\text{y}_{2}\text{ dx}\Big]$
$=2\bigg[\int\limits_{0}^{1}\sqrt{\text{x}}\text{ dx}+\int\limits_{1}^{3}\sqrt{\frac{3-\text{x}}{2}}\text{ dx}\bigg]$
$=2\Big[\frac{2}{3}\text{x}\sqrt{\text{x}}\Big]^{1}_{0}+\Big[\frac{2}{3}.\Big(\frac{3-\text{x}}{2}\Big)\sqrt{\frac{3-\text{x}}{2}}(-2)\Big]^{3}_{1} $
$=2\Big[\big(\frac{2}{3}-0\big)+(0)-\frac{2}{3}.1.1.(-2)\Big]$
$=2[\frac{2}{3}+\frac{4}{3}]$
$\text{A}=4\ \text{sq.}\ \text{units}$

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