Question

Determinar a área da região limitada pelas curvas y = x3 e y = 4x no 1º Quadrante.

Answer 1

$\mathrm{f(x)=x^3\ \ \|\ \ g(x)=4x}\\\\ \textbf{Abscissas de intersec\c{c}\~ao:}\\\\ \mathrm{f(x)=g(x)\ \to\ x^3=4x\ \to\ x^3-4x=0\ \to\ x(x^2-4)=0}\\ \mathrm{\boxed{\mathrm{x=0}}\ \ \|\ \ x^2-4=0\ \to\ x^2=4\ \to\ x=\pm\sqrt{4}\ \to\ \boxed{\mathrm{x=\pm2}}}$

$\textbf{\'Area da regi\~ao delimitada pelas curvas no 1\ºQ:}\\\\ \mathrm{S=\int_a^bg(x)-f(x)\ dx=\int_0^24x-x^3\ dx=\bigg(4\int x\ dx-\int x^3\ dx\bigg)\bigg|_0^2=}\\\\ \mathrm{=\bigg(4\dfrac{x^2}{2}-\dfrac{x^4}{4}\bigg)\bigg|_0^2=\bigg(2x^2-\dfrac{x^4}{4}\bigg)\bigg|_0^2=2.2^2-\dfrac{2^4}{4}-\bigg(2.0^2-\dfrac{0^4}{4}\bigg)=}\\\\ \mathrm{=2.4-\dfrac{16}{4}-0=8-4=4\ \to\ \boxed{\mathbf{S=4\ u.a.}}}$