Question

Determine the x-coordinate of the centroid of the shaded area.

19 — — — — — у a "І, b2 ІІ х

Answer 1

dA х X а

Equation of the bottom straight line can be written as -+-=1 a b On rearranging y=b(1-3) = (a – x) = fı(1) Rearranging the equation of upper curve y = b 1 x² b a2 az - x2 = f2(x) Now the small elemental area shown in the figure can be expressed as dA = [/2(x) - f(x)]dx Or b dA = { Vez – x2 (a – x)}dx Or da = a[/22 – x2 – a +x] dx Total area of the given figure can be computed as A= Eſ da =' } [va? –x2= a +x] dx Or -=-{\v=|dx == u de +5fr de Or

b A= a -sin -1 .x²+ b -[ax): + che le 02) - Bla?] + ba a 2 a a 0 ba (Tt - 2) 11 Centroid of the given area can be calculated as 1 b S x AA = Sx [va? – x2 = a + x]dx = -(1) The integral I can be evaluated as a a a I = x[Vaz – x2]dx ax dx + x2 dx Or a a3 1=[-(a? – x29012] [ax 2 + a3 a3 3 2 + II 3 0 Substituting this ba? 1ba3 x = -- Aa 6 6A Substituting for A ba? 2a 3(71 - 2) (ba 6 4 (11 - 2) 2a Centroid of the given area = 3- 2)