Question

∫∫x²(y-x)dxdy ,d là miền giới hạn bởi các đường y=x² và x=y²

Answer 1

It looks like the integral is

`\displaystyle \iint_D x^2 (y-x) \,\mathrm dx\,\mathrm dy`

where D is the set

D = {(x, y) : 0 ≤ x ≤ 1 and x ² ≤ y ≤ √x}

So we have

`\displaystyle \iint_D x^2(y-x)\,\mathrm dx\,\mathrm dy = \int_0^1 \int_(x^2)^(\sqrt x) x^2(y-x)\,\mathrm dy\,\mathrm dx \\\\ = \int_0^1 \left(\frac{x^2y^2}2-x^3y\right)\bigg|_(y=x^2)^(y=\sqrt x) \\\\ = \int_0^1 \left(\frac{x^3}2-x^(7/2)+x^5-\frac{x^6}2\right)\,\mathrm dx \\\\ = \left(\frac{x^4}8 - \frac{2x^(9/2)}9 + \frac{x^6}6 - (x^7)/(14)\right)\bigg|_(x=0)^(x=1) = \frac18-\frac29+\frac16-\frac1{14} = \boxed{-(1)/(504)}`