Question

The shaded area shown below is bounded by the line x - 9m on the left, the x-axis on top, and the curve y - (-18x + x2) m on the right. 9 m 18 m y=(-18x+x®) m -81 m Determine the coordinates of the centroid of the area in meters. m y = m

Answer 1

1st question:

9m Х C dy (x, y) Y = (- 18 x + x2m -81m dA= x dy centroid of strip with respect to x-axis x = 7 / y - 18 xd x² dy = (-18+ 2x) dx centroid of strip with 2 respect to y-axis with? y = y centroid of whole area with respect to x-oxis atbal} X 3x Seda Sda S (2) (scidy Sxlay >=18 X=1 2:18. SC-18+2x) dx } (x) (-18 + 2x) dx 26=9

18 S[9x² + x3] dx - 22=9 "VE 18x +2x]dx 00-9 18 х + x 4 a -18 x² 3718 2x 3 9 909 4 Cabr] - [Per +(994 4 -18 (912 +269) - 13(1034 2013 [from 2 x = 7.65m (from x= am line 7.65 + 9 = 16.65m [from y-aris j.e, x=om Centroid of whole arca y = Syda Sda worot y-axis S(3)(x) dy S (oc) dy 56-18x+x?)(3)(-18 + 2x) dx x=9 18 "S[-18 x + 2,0 2] dic x=9

18 Y = 6=26 S (324x²_S4x² + 2x 4] dx 135 [-18x + 2x] dx y [poy y sujemy + ] *), ý= - 37.26m [from from x-ascis 18 + 9 18. [chorus) 13 (+ + Coondinates of centroid of to y-axis & Y - asis are area with respect X = 16.65m = - 37.26m

2nd question:

doc IK O Х 9 (x,y) y = [.ext 57.6x]mm dA=dx centroid of strip w.rot x-axis, uc = x centroid of strip wor.t y-axis, j = 1 2 centroid of whole area wort x-oxis x= Seda SOC) (ydx) Saa Sydx 4 X = x=0 J (x) (0.9x4_ 57.6 x) dx "560.92"_57.6x]dx x=0 X 2-0 "S(0.945 - 57.6x²) dx "S(0.9x² – 57.6x]dx x=0

57.6x & Co Copy 57.640) 4. 2.22 mm 7 = Sida x = (from y-axis centroid of whole area wrot y -axis = Sy) dx Ida Sydic 7= 4 Sydx į S[0.984 – 57.66] ² dic Sydx 4 x=0 It #S(0.9x4_57.6x] dx =0 4 ý = Į (0.81x8 103.68 **+ 3317.76x3dx x=0 4 "S[0.9x4 - 57.6x] dic x=0 103.636294 4 57.6 x² y = [ 08/27 - 68(249)3317.7 Com у (from Coondinates of centroid are 2 - 42.66 mm from x-axis X = 2.22mm y = - 42.66 mm

Summary: In both the questions a strip is considered with respect to X-axis or Y-axis and corresponding centroids should be found and integrate it over entire area. Formula for centroid is mentioned in the solution, by using that formula centroid for entire area can be found.