Question

achar a area a regiao limitada pela parabola y= x^2 e y= x+2

Answer 1

$Hola! \\ \\Como ~tenemos~dos~ecuaciones, que ~son: \begin{cases}y= x^{2} \\ y=x+2\end{cases} \\ \\ Para~calcular~a~a'rea~entre ~essas~duas~func\~oes~primeiro~vamos \\ achar~a~intersec\~ao, para ~isso~igualando~as~func\~oes~temos: \\ x^{2} =x+2 \\ x^{2} -x-2=0 \\ (x-2)(x+1)=0 \\ \boxed{x_1=2~e~x_2=-1} \\ \\ Ent\~ao~devemos~calcular~a~a'rea: \\ \int_{-1} ^{2}[(x+2)- x^{2} ] dx \\ \\ \int_{-1} ^{~2} xdx+\int_{-1} ^{~2} 2dx -\int_{-1} ^{~2} x^{2} dx$
$\big[ \frac{ x^{2} }{2} +2x- \frac{ x^{3} }{3}]_{-1} ^{~2} \\ \\ \big[ \frac{ 2^{2} }{2}+2(2)- \frac{ 2^{3} }{3}]-\big[ \frac{ (-1)^{2} }{2} +2(-1)- \frac{( -1)^{3} }{3}\big] \\ \\ \frac{10}{3}+ \frac{7}{6} \\ \\ \boxed{\boxed{\frac{27}{6} ~ou~4,5 }} \\ \\ Bons~estudos!$