Answer to Question #128002 in Calculus for cherukuribindukalpana

Question #128002
determine the mass of the lamina corresponding to the first quadrant portion of the circle x^2+y^2 =25 where the density at thr point of (x,y) is f(x,y)=k*sqrt(x^2+y^2)
1
Expert's answer
2020-08-03T19:12:25-0400

"\\iint_{x^2 + y^2 \\leq 25, \\ x \\geq 0, \\ y \\geq 0} k \\sqrt{x^2 + y^2} \\,dx\\,dy ="






"= \\int_{0}^{5} \\,dx \\int_{0}^{\\sqrt{25 -x^2}} k \\sqrt{x^2 + y^2} \\,dy ="


"\\begin{cases} x = rcos\\phi \\\\ y = rsin\\phi\\\\ \\end{cases} \\\\"

"= \\int_{0}^{\\pi\/2} \\,d\\phi \\int_{0}^{5} k \\sqrt{(rcos\\phi)^2 + (rsin\\phi)^2} \\,rdr ="

"= \\int_{0}^{\\pi\/2} \\,d\\phi \\int_{0}^{5} kr^2 \\,dr = \\int_{0}^{\\pi\/2} k *5^3\/3 \\,d\\phi = k *5^3\/3 * \\pi\/2 = 125\\pi k\/6"




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