Question

Find the area of the region bounded by the two parabolas y = x2 and y2 = x.​

Answer 1

Answer:

x square and y square =x

(x+y)×(x+y)=x

x2 and y2 = x

Answer 2

${\large{\pmb{\sf{\pink{\underline{Required~ Solution}}}}}}:–$

The given two curves are parabola y = x2 and y2 = x The point of intersection of these two parabolas are 0 (0,0) and A(1,1) as shown in the fig.

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We can set

$\tt\purple{ y² = x~ or ~y = \sqrt{x}- f (x) }$

$\tt\purple{ y - x² = g(x) ~~where }$

$\tt\purple{ f(x) ≥ g(x) in [0,1]}$

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∴ The required area of the Shaded Region

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$\tt \purple{ ∫^{²}_{0}[f(x) - g(x)]dx = ∫^{l}_{0}[ \sqrt{x} - x {}^{2} ]dx }\\ \\ \\ \tt \purple{[{\frac{2}{3} {x}^{½}} = \frac{x {}^{3} }{3} ] \: = \: \frac{2}{3} \frac{ - 1}{3} = \frac{1}{3} }$

∴The Required area of the Region is 1/3 = units.