Question

1 The centroid of a surface is defined by Mox ds loyds Voz ds = y= area of o area of o area of o Find the centroid of the portion of the sphere x² + y2 + z? = 4 above the plane z = 1. Enter the exact answers. - 1 2 y II 3 =

Answer 1

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the Soint According to the given data, we have Ecuation of the Sphere as x+y+z=4 3 & also given 2=1 x + y² + 1² = 4 x+y = 3 Thus, the intersection is a disk D with radius 13 with the Symmetry principle š=ý =0 Now dv SSS E 4-3? •S da dA D SS (54-2²_y2 - 1) da D for polar Co-ordinates X=rcoso, y=rsino ours & 050 2T & xty² =22 Now s* 1984-92 - 1a.de dr.do ME 0 0

,27 6 0 0 (XXY*** - ) dr.de -> 5.de 0 lo 6 5. [at 5 5 Now May SS z dv € 4 x²-y z da da ! 4-x² - y² is - á SS (3-42-yu) da Here the polar Co-ordinates x=ncoso Yersino orso, osos27 & x + y² =12 3 (3-72) r. dr. do 27 May 0 0

122 (30-03) dr. do 0 0 Sh.de 9. [21] 0 91 le loro z = May 99/2 57/3 27 lo 2.7 Here the Symmetry principle gives that x=ý = 0 & the value of Z a 32-7 Hence, therefore Centroid is x=0 y = 0 ñ = 9.7