The area of the region bounded by parabola y2 = x and the straight line 2y = x is ______.

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The area of the region bounded by parabola y2 = x and the straight line 2y = x is $\frac{4}{3}$sq.units. Explanation: Given equation of parabola is y2 = x ......(i) And equation of straight line is 2y = x ......(ii) Solving equation (i) and (ii) We get (x/2)^2 = x $\Rightarrow$ x^$\frac{2}{4}$ = x $\Rightarrow$ x2 = 4x $\Rightarrow$ x(x – 4) = 0 $\therefore$ x = 0, 4 Required area = `int_0^4 sqrt(x) "d"x - int_0^4 x/2 "d"x = $\frac{2}{3}$ [x^($\frac{3}{2}$)]_0^4 - $\frac{1}{2}$ * $\frac{1}{2}$ [x^2]_0^4 = $\frac{2}{3}$ [(4)^($\frac{3}{2}$) - 0] - $\frac{1}{4}$ [(4)^2 - 0] = $\frac{2}{3}$ xx 8 - $\frac{1}{4}$ xx 16 = $\frac{16}{3}$ - 4 = $\frac{4}{3}$` sq.units [Solution diagram: 251740_solution_1.png]

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