Area bounded by parabola y2 = x and straight line 2y = x is _________ .

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KollegeApply · Accepted Answer

Point of intersection is obtained by solving the equation of parabola y2 = x and equation of line 2y= x, we have y^2 = x\text{ and }2y = x \Rightarrow y^2 = 2y \Rightarrow y^2 - 2y = 0 \Rightarrow y = 0\text{ or }y = 2 \Rightarrow x = 0\text{ or }x = 4 \text{ Thus O }\left( 0, 0 \right)\text{ and A }\left( 4, 2 \right) \text{ are the points of intersection of the curve and straight line } . Area bound by them] A = \int_0^4 \left( y_1 - y_2 \right) dx .............\left[\text{Where, }y_1 = $\sqrt$(x)\text{ and }y_2 = x/2 \right] = \int_0^4 \left( $\sqrt$(x) - x/2 \right) dx = \left[ x^3/2/3/2 - $\frac{1}{2}$ $\times$ x^$\frac{2}{2}$ \right]_0^4 = \left[ $\frac{2}{3}$ x^$\frac{3}{2}$ - x^$\frac{2}{4}$ \right]_0^4 = $\frac{2}{3}$ 4^$\frac{3}{2}$ - $\frac{1}{4}$ $\times$ 4^2 - 0 = $\frac{2}{3}$ $\times$ 2^3 - $\frac{16}{4}$ = $\frac{16}{3}$ - 4 = 16 - $\frac{12}{3}$ = $\frac{4}{3}$\text{ sq units } [Solution diagram: 44105_solution_1.png]

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