Area Question 20
Question 20
- The area (in sq units) bounded by the curves $y=\sqrt{x}, 2 y-x+3=0, X$-axis and lying in the first quadrant, is
(2013 Main, 03) (a) 9 (b) 6 (c) 18 (d) $\frac{27}{4}$
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Solution:
- Given curves are $y=\sqrt{x}$
and
$2 y-x+3=0$
On solving Eqs. (i) and (ii), we get
$$ \begin{array}{rlrl} & & 2 \sqrt{x}-(\sqrt{x})^{2}+3 & =0 \ \Rightarrow \quad & (\sqrt{x})^{2}-2 \sqrt{x}-3 & =0 \ \Rightarrow \quad & & (\sqrt{x}-3)(\sqrt{x}+1) & =0 \quad \Rightarrow \quad \sqrt{x}=3 \ & & & \ \therefore & & & =3 \end{array} $$
Hence, required area
$$ \begin{aligned} & =\int_{0}^{3}\left(x_{2}-x_{1}\right) d y=\int_{0}^{3}\left{(2 y+3)-y^{2}\right} d y \ & =y^{2}+3 y-{\frac{y^{3}}{3}}_{3}^{3}=9+9-9=9 \text { sq units } \end{aligned} $$
