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A lamina occupies the part of the disk x2+y2≤4 in the top half. Find its mass if the density at any point is 2.3 times the point's distance from the origin.

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Answer: Its mass is 9.2 if the density at any point is 2.3 times the point's distance from the origin.

Explanation:

Since we have given that


x2+y2\leq 4

Take polar coordinate:


x=r\cos \theta\\\\y=r\sin \theta

And we know that


x^2+y^2=r^2\\\\dA=rdrd\theta

So, it becomes,


0\leq r^2\leq 4\\\\0\leq r\leq 2

Since density at any points is 2.3 times the point's distance from the origin.


\rho=2.3(√(x^2+y^2))^2\\\\\rho=2.3(x^2+y^2)\\\\\rho=2.3r^2

So, the mass of the lamina is given by


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)* [(r^4)/(4)]_2^0\\\\m=(2.3)/(4)(2\pi-\pi)(16-0)\\\\m=(2.3\pi)/(4)* 16\\\\m=2.3* 4=9.2

Hence, its mass is 9.2 if the density at any point is 2.3 times the point's distance from the origin.

User Bob Desaunois
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