Respuesta :
Answer: Its mass is 9.2 if the density at any point is 2.3 times the point's distance from the origin.
Step-by-step explanation:
Since we have given that
[tex]x2+y2\leq 4[/tex]
Take polar coordinate:
[tex]x=r\cos \theta\\\\y=r\sin \theta[/tex]
And we know that
[tex]x^2+y^2=r^2\\\\dA=rdrd\theta[/tex]
So, it becomes,
[tex]0\leq r^2\leq 4\\\\0\leq r\leq 2[/tex]
Since density at any points is 2.3 times the point's distance from the origin.
[tex]\rho=2.3(\sqrt{x^2+y^2})^2\\\\\rho=2.3(x^2+y^2)\\\\\rho=2.3r^2[/tex]
So, the mass of the lamina is given by
[tex]m=\int\limits^a_b \int\limits^a_b {\rho} dA\\\\m=\int\limits^{2\pi }_{\pi}\int\limits^2_0 {2.3r^2.r} \, drd\theta\\\\m=2.3\int\limits^{2\pi }_{\pi} \, d\theta\int\limits^2_0 {r^3} \, dr\\\\m=2.3[\theta]_{\pi}^{2\pi}\times [\dfrac{r^4}{4}]_2^0\\\\m=\dfrac{2.3}{4}(2\pi-\pi)(16-0)\\\\m=\dfrac{2.3\pi}{4}\times 16\\\\m=2.3\times 4=9.2[/tex]
Hence, its mass is 9.2 if the density at any point is 2.3 times the point's distance from the origin.
