Question

AM 232 politehnica 2019:Sa se determine aria domeniului mărginit,delimitat de parabola y^2 =10 si dreapta y=5x.
Raspunsuri:a 3/2;b 4/3;c 9/2 ;d2/15; e1/3; f 1/5​

Answer 1

$y^2=10x\\ y = 5x \\\\x = \dfrac{y^2}{10} \Rightarrow f(y) = \dfrac{y^2}{10}\\ \\x = \dfrac{y}{5}\Rightarrow g(y) = \dfrac{y}{5} \\ \\ f(y)=g(y) \Rightarrow \dfrac{y^2}{10} = \dfrac{y}{5} \Rightarrow y^2 = 2y \Rightarrow y(y-2) = 0 \Rightarrow \\\\\Rightarrow y_1=0,y_2=2 \Rightarrow \text{Intervalul de integrare este: }[0,2] \\ \\ \Rightarrow A = \displaystyle |\int_{y_1}^{y_2}\Big(f(y)-g(y)\Big)\, dy | =$

$\displaystyle = |\int_{0}^{2}\Big(\dfrac{y^2}{10}-\dfrac{y}{5}\Big)\, dy |= |\Big(\dfrac{y^3}{30}-\dfrac{y^2}{10}\Big)\Big|_0^2 | = \\ \\ =|\dfrac{8}{30}-\dfrac{4}{10}| =|\dfrac{4}{15}-\dfrac{2}{5}| = |\dfrac{4-6}{15} = |-\dfrac{2}{15}| = \boxed{\dfrac{2}{15}}$