Question

Buna ziua,


Ma puteti ajuta cu acest exercitiu?

ffD xydxdy, unde D domeniul marginit de curbele de ecuatii y=x la puterea a doua si ay-bx=0 , unde a=5, b=4

Answer 1

Răspuns:

I=∫∫Dxydxdy

y=x²

ay-bx=0

Faci inlocuirile

5y-4x=0

Determini limitele de integrare

5*x²-4x=0

x(5x-4)=0

x1=0

x2=5/4

Observi  conf regulii semnelr pt functia de gradul 2 ca expresia e negativa (esti intre radacini)=> dreapa 5y-4x=0 este deasupra parabolei y=x²

I=∫∫xydx dy=

∫($\int\limits^\frac{5x}{4} _ {x^2} \,xy dy))dx$

$\int\limits^\frac{5x}{4} _ {x^2} xy\, dx =$

$x\int\limits^\frac{5x}{4} _{x^2} \,y dx =$

x$\frac{y^2}{2}$║⁵ˣ/4ₓ²=

x*$(\frac{5x}{4} )^2-x^2)$=

x($\frac{25x^2}{16} -x^2)$

$x\frac{9x^2}{16}$=

$\frac{9x^3}{16} =$

I=$\int\limits^\frac{5}{4} _ {0} \,\frac{9x^3}{16} dx =$

$\frac{9}{16} \int\limits^\frac{5}{4} _ {0} \,x^3 dx$

$\frac{9}{16} \frac{x^4}{4}$║₀⁵/4=

$\frac{9}{16} *[(\frac{5}{4} )^4-0]$

$\frac{9}{4^2} *\frac{5^4}{4^4}$

$\frac{9*625}{4^6}$

Explicație pas cu pas: