Question
Find tha area bounded by the curves x = y2 and x = 3 - 2y2.
✓
Answer
To find area bounded by
x = y2 ...(i)
and x = 3 - 2y2
2y2 = -(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 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} $
$=2\Big[\big(\frac{2}{3}-0\big)+(0)-\frac{2}{3}.1.1.(-2)\Big]$
$=2[\frac{2}{3}+\frac{4}{3}]$
$\text{A}=4\ \text{sq.}\ \text{units}$
x = y2 ...(i)
and x = 3 - 2y2
2y2 = -(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 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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