Question

Determine o momento de inercia Ix e Iy da região limitada pelos gráficos y = x^3 e y = 2x. Onde a densidade é d(x,y) = x. Utilizando integral dupla.

Answer 1

Seja $A=A_1\cup A_2$ onde
 
  $A_1=\left\{(x,y): -\sqrt{2}\leq x\leq 0\,,\,2x\leq y\leq x^3\right\}\\ \\ A_2=\left\{(x,y): 0\leq x\leq \sqrt{2}\,,\,x^3\leq y\leq 2x\right\}\\ \\ \\ \text{Momentos de Inercia}\\ \\ \displaystyle I_x=\iint y^2\rho(x,y)\,dA\\ \\ I_x=\iint y^2\rho(x,y)\,dA_1+\iint y^2\rho(x,y)\,dA_2\\ \\ I_x=\iint xy^2\,dA_1+\iint xy^2\,dA_2$
 
   $\displaystyle I_x=\int\limits_{-\sqrt{2}}^{0}\int\limits_{2x}^{x^3}xy^2\; dy\,dx+\int\limits_{0}^{\sqrt{2}}\int\limits_{x^3}^{2x}xy^2\; dy\,dx \\ \\ \\ I_x=\int\limits_{-\sqrt{2}}^{0}x\int\limits_{2x}^{x^3}y^2\; dy\,dx+\int\limits_{0}^{\sqrt{2}}x\int\limits_{x^3}^{2x}y^2\; dy\,dx \\ \\ \\ I_x=\dfrac{1}{3}\int\limits_{-\sqrt{2}}^{0}x^{10}-8x^4\,dx+\dfrac{1}{3}\int\limits_{0}^{\sqrt{2}}8x^4-x^{10}\, dx\\ \\ \\ I_x=-\dfrac{192\sqrt{2}}{55}+\dfrac{192\sqrt{2}}{55}\\ \\ \\ \boxed{I_x=0}$