Question

Sa se calculeze ∫∫ D ydxdy unde D este limitat de y=x²si y=2x.

Answer 1

$\displaystyle\\D:\begin{cases}x\in [0,2]\\ y\in [x^2,2x]\end{cases}.\\\iint_Dy~\textnormal{dxdy}=\int_0^2\int_{x^2}^{2x} y~\textnormal{dydx}=\int_0^2\left(\int_{x^2}^{2x}y~\textnormal{dy}\right)~\textnormal{dx}=\int_0^2 2x^2-\frac{x^4}{2}~\textnormal{dx}=\\\left[\frac{2x^3}{3}-\frac{x^5}{10}\right]_0^2=\frac{32}{15}.$