Answer to Question #142086 in Calculus for Promise Omiponle

Question #142086
Find the centroid (x¯,y¯) of the region bounded by:
y=7x^2+6x, y=0, x=0, and x=6.
1
Expert's answer
2020-11-09T19:43:12-0500
SolutionSolution

Given y=7x2+6x,y=0,x=0,x=6y=7x^2+6x,y=0,x=0,x=6




dA=7x2+6xxˉ=06xdA06dA=xˉ=06x(7x2+6x)dx06(7x2+6x)dx=067x3dx+066xdx067x2dx+066xdx=[7x44]06+[6x22]06[7x33]06+[6x22]06=[74(640)]+[62(640)][73(640)]+[62(640)]xˉ=2268+108504+108=2376612=3.88dA=7x^2+6x\\ \bar x=\frac{\intop_0^6 x dA}{\intop_0^6 dA}=\bar x=\frac{\intop_0^6 x (7x^2+6x)dx}{\intop_0^6 (7x^2+6x)dx}\\ =\frac{\intop_0^6 7x^3dx+\intop_0^6 6xdx}{\intop_0^6 7x^2dx+\intop_0^66xdx}=\frac{[\frac{7x^4}{4}]_0^6+[\frac{6x^2}{2}]_0^6}{[\frac{7x^3}{3}]_0^6+[\frac{6x^2}{2}]_0^6}=\frac{[\frac{7}{4}(6^4-0)]+[\frac{6}{2}(6^4-0)]}{[\frac{7}{3}(6^4-0)]+[\frac{6}{2}(6^4-0)]}\\ \bar x=\frac{2268+108}{504+108}=\frac{2376}{612}=3.88\\

yˉ=06ydA06dA=yˉ=06(7x2+6x)2dx06(7x2+6x)dx=0649x4dx+84x3+36x2067x2dx+066xdx=49(x55)06+84(x44)06+36(x33)06[7x33]06+[6x22]06=49(655)+84(644)+36(633)[7(6)33]+[6(6)22]yˉ=76204.8+27216+2592504+108=106012.8612=1732.23(xˉ,yˉ)=(3.88,1732.23)\bar y=\frac{\intop_0^6 y dA}{\intop_0^6 dA}=\bar y=\frac{\intop_0^6 (7x^2+6x)^2dx}{\intop_0^6 (7x^2+6x)dx}\\ =\frac{\intop_0^6 49x^4dx+84x^3+36x^2}{\intop_0^6 7x^2dx+\intop_0^66xdx}=\frac{49(\frac{x^5}{5})_0^6+84(\frac{x^4}{4})_0^6+36(\frac{x^3}{3})_0^6}{[\frac{7x^3}{3}]_0^6+[\frac{6x^2}{2}]_0^6}=\frac{49(\frac{6^5}{5})+84(\frac{6^4}{4})+36(\frac{6^3}{3})}{[\frac{7(6)^3}{3}]+[\frac{6(6)^2}{2}]}\\ \bar y=\frac{76204.8+27216+2592}{504+108}=\frac{106012.8}{612}=1732.23\\ \therefore (\bar x, \bar y)=(3.88, 1732.23)




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