Question

Find the area bounded by the curve using integration

$\frac{ {x}^{2} }{9} + \frac{ {y}^{2} }{4} = 1$
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Answer 1

$\large\underline{\sf{Solution-}}$

Given curve is

$\rm \: \dfrac{ {x}^{2} }{9} + \dfrac{ {y}^{2} }{4} = 1$

It represents the equation of ellipse having center (0, 0) and intersecting the x - axis at (3, 0) and (- 3, 0) and y - axis at (0, 2) and (0, - 2).

Now,

$\rm \:\dfrac{ {y}^{2} }{4} = 1 - \dfrac{ {x}^{2} }{9}$

$\rm \:\dfrac{ {y}^{2} }{4} = \dfrac{9 - {x}^{2} }{9}$

$\rm \: {y}^{2} =4\bigg( \dfrac{9 - {x}^{2} }{9} \bigg)$

$\rm\implies \:y = \dfrac{2}{3} \sqrt{ {3}^{2} - {x}^{2} }$

Now, as ellipse is symmetrical in all the four quadrants. So, we find the area of ellipse in first quadrant with respect to x axis from x = 0 to x = 3 and to find the total area, we multiply by 4.

So, required area is

$\rm \: = \: 4\displaystyle\int_{0}^{3}\rm y \: dx$

$\rm \: = \: 4 \times \frac{2}{3} \displaystyle\int_{0}^{3}\rm \sqrt{ {3}^{2} - {x}^{2} } \: dx$

$\rm \: = \: \dfrac{8}{3}\bigg(\dfrac{x}{2} \sqrt{ {3}^{2} - {x}^{2} } + \dfrac{ {3}^{2} }{2} {sin}^{ - 1} \dfrac{x}{3} \bigg)_{0}^{3}$

$\rm \: = \: \dfrac{8}{3}\bigg(0+ \dfrac{ {3}^{2} }{2} {sin}^{ - 1} \dfrac{3}{3} - 0 - 0\bigg)$

$\rm \: = \: \dfrac{8}{3}\bigg(\dfrac{9}{2} {sin}^{ - 1}1\bigg)$

$\rm \: = \: 12 \times \frac{\pi}{2}$

$\rm \: = \: 6\pi \: square \: units$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ {x}^{2} + {a}^{2} } = \dfrac{1}{a} {tan}^{ - 1} \dfrac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log |x + \sqrt{ {x}^{2} + {a}^{2}} | + c}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$