Question

encontre a area da figura englobada pelas curvas y= x^2 e y=x+6

Answer 1

Boa noite Marcelo!

Solução!

Vamos aplicar uma integral em que os limites de integração são x=-2 e x=3.

$A=\int\limits^3_ {-2}[(x+6)] - x^{2} dx=\\\\\\\\ A=[ \frac{ x^{2} }{2} +6x- \frac{ x^{3} }{3}]_{-2} ^{3}=\\\\\\\ A=[ (\dfrac{ 9 }{2} +18- \frac{ 27 }{3})- (\frac{ -2^{2} }{2} -12+ \frac{ 8 }{3})]\\\\\\\ A=[ (\dfrac{ 9 }{2} +18- 9)- (\frac{ 4 }{2} -12+ \frac{ 8 }{3})]\\\\\\\\\ A=[ (\dfrac{ 9 }{2} +18- 9)- (2 -12+ \frac{ 8 }{3})]\\\\\\\\\ A=[ (\dfrac{ 9+36-18 }{2} )- ( \frac{6-36+ 8 }{3})]\\\\\\\\\ A=[ (\dfrac{ 27 }{2} )- ( \frac{-22 }{3})]\\\\\\\ A=[ (\dfrac{ 27 }{2} + \frac{22 }{3})]$

$A= \dfrac{81+44}{6} \\\\\ A= \dfrac{125}{6}$




$\boxed{Resposta:A= \frac{125}{6}}$

Boa noite!
Bons estudos!