Question

1. Calcule a área de região delimitada pelos gráficos das funções f(x) = 3x e g(x) = x^2

Answer 1

Olá


$\displaystyle\mathsf{Area= \int\limits^b_a {f(x)-g(x)} \, dx }$



Vamos encontrar os limites de integração.


f(x) = g(x)

3x = x²


3x - x² = 0

x(3 - x) = 0


x = 0   ,     x = 3                        ←      Esses são os limites de integração



Portanto



$\displaystyle\mathsf{A= \int\limits^3_0 {(3x-x^2)} \, dx }\\\\\\\\\mathsf{A={\left( \frac{3x^{1+1}}{1+1} - \frac{x^{2+1}}{2+1} \right)\bigg|^3_0} \, }\\\\\\\\\mathsf{A={\left( \frac{3x^{2}}{2} - \frac{x^{3}}{3} \right)\bigg|^3_0} \, }\\\\\\\\\mathsf{A={\left( \frac{3(3)^{2}}{2} - \frac{(3)^{3}}{3} \right)~-~\left( \frac{3(0)^{2}}{2} - \frac{(0)^{3}}{3} \right)} \, }\\\\\\\\\mathsf{A={\left( \frac{27}{2} - \frac{27}{3} \right)~-~ -0}}\\\\\\\\\mathsf{A={ \frac{81-54}{6} }}$


$\displaystyle \mathsf{A={ \frac{27^{\div 3}}{6^{\div 3}} }}\\\\\\\boxed{\mathsf{A= \frac{9}{2} ~u.a.}}$




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